Answer:
10
Step-by-step explanation:
The number of tiles in the design is 1 + 2 + 3 + ...
We can model this as an arithmetic series, where the first term is 1 and the common difference is 1. The sum of the first n terms of an arithmetic series is:
S = n/2 (2a₁ + d (n − 1))
Given that a₁ = 1 and d = 1:
S = n/2 (2(1) + n − 1)
S = n/2 (n + 1)
Since S ≤ 60:
n/2 (n + 1) ≤ 60
n (n + 1) ≤ 120
n must be an integer, so from trial and error:
n ≤ 10
Mr. Tong should use 10 tiles in the final row to use the most tiles possible.
Suppose that vehicles taking a particular freeway exit can turnright (R), turn left (L), or go straight (S). Consider observingthe direction for each of three successive vehicles.
a. List all outcomes in the event A that all three vehicles go inthe same direction.
b. List all outcomes in the event B that all three vehicles takedifferent directions.
c. list all outcomes in the event C that exactly two of the threevehicles turn right.
d. List all outcomes in the event C that exactly two of the threevehicles turn right.
e. List outcomes in D', C U D, and C D
Answer:
a.) A= [RRR, LLL, SSS]
b.) B=[RLS, RSL, SRL, SLR, LRS, LSR]
c.) C=[RRL, RRS, RSR, RLR, SRR, LRR]
e.) D'= C' = [RRR, LLL, SSS, RLS, RSL, SRL, SLR, LRS, LSR, LLR, LLS, LSL, LRL, SLL, RLL, SSL, SSR, SLS, SRS, RSS, LSS]
Step-by-step explanation:
If three consecutive vehicles are considered with their direction as R, L and S, then
a.) when the three vehicles go in same direction, A= [RRR, LLL, SSS]
b.) When the three vehicles take the different direction, B=[RLS, RSL, SRL, SLR, LRS, LSR]
c.) when exactly two vehicles turn right, C=[RRL, RRS, RSR, RLR, SRR, LRR]
d.) repeated question of C above.
e.) D'= C' = [RRR, LLL, SSS, RLS, RSL, SRL, SLR, LRS, LSR, LLR, LLS, LSL, LRL, SLL, RLL, SSL, SSR, SLS, SRS, RSS, LSS]
What are some solutions to nonresponse? Select all that apply. A. reduce undercoverage B. use stratified sampling C. use convenience sampling D. change wording of questions E. offer rewards and incentives F. reduce interview error G. attempt callbacks H. use cluster sampling
A non responses is a failure to reply something and is a condition that is not responding.
There exists various factors that can create this effect, for example: type of survey, bad questions, un-probabilistic sample, etc.
By the offering of rewards and the incentives It is true as people get a reward or the incentive they would be more willing to rely. A reduce interview error is False as the interview error is not directly linked to the non response bias .Hence the options E and F are correct.
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To tackle nonresponse in surveys, strategies such as reducing undercoverage, using stratified sampling, changing the wording of questions, offering incentives, reducing interview error, and attempting callbacks can be effective. These methods help enhance response rates and the reliability of survey data.
Tackling nonresponse in surveys is crucial for ensuring accurate and reliable data. Here are some effective solutions:
→ Reduce Undercoverage: By ensuring the survey reaches all relevant subpopulations, you can minimize the chances of missing out on certain groups.
→ Use Stratified Sampling: This method can enhance response rates by making sure each subgroup is adequately represented.
→ Change Wording of Questions: Making questions clearer and more straightforward can increase the likelihood of responses.
→ Offer Rewards and Incentives: Providing incentives can motivate participants to complete the survey.
→ Reduce Interview Error: Training interviewers to minimize bias and errors can improve response quality.
→ Attempt Callbacks: Following up with nonrespondents can help in obtaining more responses.
These methods are essential to improve response rates and, consequently, the accuracy of survey results.
How many observations should be made if she wants to be 86.64 percent confident that the maximum error in the observed time is .5 second? Assume that the standard deviation of the task time is four seconds.
Answer:
144 observations
Step-by-step explanation:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1-0.8684}{2} = 0.0668[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].
So it is z with a pvalue of [tex]1-0.0668 = 0.9332[/tex], so [tex]z = 1.5[/tex]
Now, find the margin of error M as such
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
In this problem, we have that:
[tex]\sigma = 4, M = 0.5[/tex]
We want to find n
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
[tex]0.5 = 1.5*\frac{4}{\sqrt{n}}[/tex]
[tex]0.5\sqrt{n} = 6[/tex]
[tex]\sqrt{n} = \frac{6}{0.5}[/tex]
[tex]\sqrt{n} = 12[/tex]
[tex]\sqrt{n}^{2} = (12)^{2}[/tex]
[tex]n = 144[/tex]
Express the negations of each of these statements so that all negation symbols immediately precede predicates. a. ∃z∀y∀xT (x, y, z) b. ∃x∃yP (x, y) ∧ ∀x∀yQ(x, y) c. ∃x∃y(Q(x, y) ↔ Q(y, x)) d. ∀y∃x∃z(T (x, y, z) ∨ Q(x, y))
Final answer:
The negations of the given predicate logic statements are expressed by switching the quantifiers (∀ and ∃) and adding negation symbols (¬) directly before the predicates, resulting in new statements that oppose the original ones.
Explanation:
The goal is to express the negation of each of the given predicate logic statements such that all negation symbols immediately precede predicates.
For the statement ∃z∀y∀xT(x, y, z), the negation would be ∀z∃y∃x¬T(x, y, z), which states that there is no z for which every y and every x make T(x, y, z) true.The negation of ∃x∃yP(x, y) ∧ ∀x∀yQ(x, y) is ∀x∀y¬P(x, y) ∨ ∃x∃y¬Q(x, y), meaning there are no such x and y that P(x, y) is true or there exists some x and y for which Q(x, y) is not true.For ∃x∃y(Q(x, y) ↔ Q(y, x)), the negation would be ∀x∀y¬(Q(x, y) ↔ Q(y, x)), indicating that for all x and y, it is not the case that Q(x, y) if and only if Q(y, x).The negation of the statement ∀y∃x∃z(T(x, y, z) ∨ Q(x, y)) is ∃y∀x∀z(¬T(x, y, z) ∧ ¬Q(x, y)), stating there exists a y such that for all x and z, neither T(x, y, z) nor Q(x, y) are true.device uses five silicon chips. Suppose the five chips are chosen at random from a batch of a hundred chips out of which five are defective. What is the probability that the de\"ice contains no defecth'e chip when it is made up from one batch?
The probability that a device using five silicon chips selected randomly from a batch of 100 chips, which includes five defective ones, contains no defective chip is calculated by the ratio of selecting five good chips to selecting any five chips from the batch.
Explanation:The question is asking to find the probability that a device, which uses five silicon chips selected from a batch of a hundred chips with five being defective, will have no defective chip. To solve this, we can calculate the probability step by step using the concept of combinations.
Firstly, we determine the number of ways to select five non-defective chips out of 95 good ones, which is C(95,5). Then, we calculate the total number of ways to select any five chips out of the whole batch, which is C(100,5). The probability that the device contains no defective chip is the ratio of these two numbers:
P(device has no defective chip) = C(95,5) / C(100,5)
Where C(n,k) represents the number of combinations of n items taken k at a time.
To calculate this, use factorials where C(n,k) = n! / [(n-k)!k!].
So, the probability that the device contains no defective chip, is:
P(device has no defective chip) = (95! / (90!*5!)) / (100! / (95!*5!))
Simplifying the factorials, we have:
P(device has no defective chip) = (95*94*93*92*91) / (100*99*98*97*96)
Finally, calculate this to get the decimal value, which would give the probability that the device contains no defective chips when made up from one batch.
Determine whether the given value is a discrete or continuous variable. People are asked to state how many times in the last month they visited their family doctor.
Discrete
Continuous
Answer:
The given value is discrete variable.
Step-by-step explanation:
Discrete Variable:
Discrete Variable are those variables that can only take on a finite number of values are called "discrete variables." All qualitative variables are discrete. Some quantitative variables are discrete, such as performance rated as 1,2,3,4, or 5, or temperature rounded to the nearest degree.
Here They have visited the doctor many times so it will be a whole number for sure.
According to a survey of adults, 64 percent have money in a bank savings account. If we were to survey 50 randomly selected adults, find the mean number of adults who would have bank savings accounts.
Answer:
The mean number of adults who would have bank savings accounts is 32.
Step-by-step explanation:
For each adult surveyed, there are only two possible outcomes. Either they have bank savings accounts, or they do not. So we use the binomial probability distribution to solve this problem.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
In this problem, we have that:
[tex]p = 0.64[/tex]
If we were to survey 50 randomly selected adults, find the mean number of adults who would have bank savings accounts.
This is E(X) when [tex]n = 50[/tex].
So
[tex]E(X) = np = 50*0.64 = 32[/tex]
The mean number of adults who would have bank savings accounts is 32.
Using the extended Euclidean algorithm, find the multiplicative inverse of a. 1234 mod 4321 b. 24140 mod 40902
(a) The inverse of 1234 (mod 4321) is x such that 1234*x ≡ 1 (mod 4321). Apply Euclid's algorithm:
4321 = 1234 * 3 + 619
1234 = 619 * 1 + 615
619 = 615 * 1 + 4
615 = 4 * 153 + 3
4 = 3 * 1 + 1
Now write 1 as a linear combination of 4321 and 1234:
1 = 4 - 3
1 = 4 - (615 - 4 * 153) = 4 * 154 - 615
1 = 619 * 154 - 155 * (1234 - 619) = 619 * 309 - 155 * 1234
1 = (4321 - 1234 * 3) * 309 - 155 * 1234 = 4321 * 309 - 1082 * 1234
Reducing this leaves us with
1 ≡ -1082 * 1234 (mod 4321)
and so the inverse is
-1082 ≡ 3239 (mod 4321)
(b) Both 24140 and 40902 are even, so there GCD can't possibly be 1 and there is no inverse.
The multiplicative inverse of a number is simply its reciprocal
The multiplicative inverse of 1234 mod 4321 is [tex]\mathbf{ -1082 \equiv 3239\ (mod\ 4321)}[/tex].24140 mod 40902 as no multiplicative inverse.To determine the multiplicative inverse of a mod b, one of a and b must not be an even number
(a) Multiplicative inverse of 1234 mod 4321
This can be written as:
[tex]\mathbf{1234 \times x \equiv 1\ (mod\ 4321)}[/tex]
When the extended Euclidean's algorithm is applied, we start by writing the expression in the following format:
[tex]\mathbf{Dividend = Quotient \times Divisor + Remainder}[/tex]
So, we have:
[tex]\mathbf{4321 = 1234 \times 3 + 619}[/tex]
Express 1234 using the above format
[tex]\mathbf{1234 = 619 \times 1 + 615}[/tex]
Repeat the process for all quotient
[tex]\mathbf{619 = 615 \times 1 + 4}[/tex]
[tex]\mathbf{615 = 4 \times 153 + 3}[/tex]
[tex]\mathbf{4= 3 \times 1 + 1}[/tex]
Next, we reverse the process as follows:
Make 1 the subject in [tex]\mathbf{4= 3 \times 1 + 1}[/tex]
[tex]\mathbf{1 = 4 - 3}[/tex]
Substitute an equivalent expression for 3
[tex]\mathbf{1 = 4 - (615 - 4 \times 153)}[/tex]
[tex]\mathbf{1 = 4 - 615 + 4 \times 153}[/tex]
Collect like terms
[tex]\mathbf{1 = 4 + 4 \times 153 - 615 }[/tex]
[tex]\mathbf{1 = 4 \times 154 - 615 }[/tex]
Substitute an equivalent expression for 615
[tex]\mathbf{1 = 619 \times 154 - 155 \times (1234 - 619) }[/tex]
[tex]\mathbf{1 = 619 \times 309 - 155 \times 1234 }[/tex]
Substitute an equivalent expression for 619
[tex]\mathbf{1 =(4321 - 1234 \times 3) \times 309 - 155 \times 1234}[/tex]
[tex]\mathbf{1 = 4321 \times 309 - 1082 \times 1234}[/tex]
Recall that:
[tex]\mathbf{1234 \times x \equiv 1\ (mod\ 4321)}[/tex]
So, we have:
[tex]\mathbf{1 \equiv -1082 \times 1234\ mod(4321)}[/tex]
Add 4321 and -1082
[tex]\mathbf{4321 -1082 = 3239}[/tex]
Hence, the required inverse is:
[tex]\mathbf{ -1082 \equiv 3239\ (mod\ 4321)}[/tex]
(b) Multiplicative inverse of 24140 mod 40902
Recall that:
To determine the multiplicative inverse of a mod b, one of a and b must not be an even number
Because 24140 and 40902 are both even numbers, then:
24140 mod 40902 has no multiplicative inverse
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The number of contaminating particles on a silicon wafer prior to a certain rinsing process was determined for each wafer in a sample of size 100, resulting in the following frequencies:
Number of particles: 0, 1, 2, 3, 4, , 5, 6, 7
Frequency: 1, 2, 3, 12, 11, 15, 18, 10
Number of particles: 8, 9, 10, 11, 12, 13, 14
Frequency: 12, 4, 5, 3, 1, 2, 1
(a.) What proportion of the sampled wafers had at least one particle? At least five particles?
(b.) What proportion of the sampled wafers had between five and ten particles, inclusive? Strictly between five and ten particles?
(c.) Draw a histogram using relative frequency on the vertical axis. How would you describe the shape of the histogram.
The proportion of sample wafers that had at least one particle is 96%, those with at least five particles is 71%. For wafers between five and ten particles (inclusive), the proportion is 64%, and strictly between five and ten particles it is 44%. The histogram based on the given data would appear roughly bell-shaped with a right skew.
Explanation:(a.) To calculate the proportion of sampled wafers that had at least one particle, we sum the frequencies of all groups with one or more particles and divide by the total sample size. So, the proportion is given by: (2+3+12+11+15+18+10+12+4+5+3+1+2+1)/100 = 96/100 = 0.96 or 96%.
Similarly, for wafers with at least five particles, the calculation is: (15+18+10+12+4+5+3+1+2+1)/100 = 71/100 = 0.71 or 71%.
(b.) For wafers with between five and ten particles (inclusive), we sum frequencies from 5 to 10 particles, giving (15+18+10+12+4+5)/100 = 64/100 = 0.64 or 64%.
For wafers strictly between five and ten particles (i.e., 6 to 9 particles inclusive), the calculation is: (18+10+12+4)/100 = 44/100 = 0.44 or 44%.
(c.) A histogram with relative frequency would show the proportion of wafers (y-axis) against the number of particles (x-axis). Given the distribution of frequencies, it might be expected that the histogram appears roughly bell-shaped, though the peak would be skewed to the right considering higher numbers around 5 to 8 particles.
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Find the balance of $7,000 deposited at 4% compounded semi-annually for 2 years
Answer:
The balance will be $7,577.03.
Step-by-step explanation:
The compound interest formula is given by:
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
Where A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.
In this problem, we have that:
[tex]P = 7000, r = 0.04[/tex]
Semianually means twice a year, so [tex]n = 2[/tex]
We want to find A when [tex]t = 2[/tex].
So
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
[tex]A = 7000(1 + \frac{0.04}{2})^{2*2}[/tex]
[tex]A = 7577.03[/tex]
The balance will be $7,577.03.
The data below represent the top speed (in kilometers per hour) of all the players (except goaltenders) in a certain soccer league.
Number of Players 5 Speed (km/hr) 10-13.9 14-17.9 18-21.9 22-25.9 26-29.9 30-33.9 16 80 259
Find (a) the number of classes, (b) the class limits for the third class, and (c) the class width
Answer:
a. 6
b. 18-21.9
c. 4
Step-by-step explanation:
a)
The number of classes can be assessed by counting the number of class intervals
S.no Class interval
1 10-13.9
2 14-17.9
3 18-21.9
4 22-25.9
5 26-29.9
6 30-33.9
Hence, there are 6 number of classes
b)
The class limits for third class is 18-21.9.
Upper class limit= 18
Lower class limit=21.9
c)
The class width can be calculated by taking difference of two consecutive upper class limits or two consecutive lower class limits.
Class width=14-10=4
The data represent the top speeds of players in a soccer league with 7 classes, the class limits for the third class are 18 to 21.9 km/hr, and the class width is 3.9 km/hr.
Explanation:The data provided represents the top speed of soccer players in specific intervals known as classes. We can analyze the information to answer the student's queries as follows:
Number of classes: This is the number of intervals used to categorize the top speed. Simply count the number of intervals.Class limits for the third class: These are the boundaries of the third speed interval. Look at the lower and upper bounds of the third interval.Class width: This is the difference between the upper and lower class limits. Calculate it by subtracting the lower limit of any class from its upper limit.Based on the data provided, if we assume that the sequence goes like 10-13.9, 14-17.9, 18-21.9, and so on, we can derive the following:
The number of classes is 7, since there are 7 speed intervals.The class limits for the third class are 18 km/hr (lower limit) and 21.9 km/hr (upper limit).The class width can be calculated by taking any interval, for example, 14-17.9. Subtracting 14 from 17.9 gives us a class width of 3.9 km/hr.Find two vectors in opposite directions that are orthogonal to the vector u. (The answers are not unique. Enter your answer as a comma-separated list of vectors.) u = 5, −2, 8
The two vectors which are orthogonal to the given vector u are (0, 4, 1) and (8, 0, -5).
Use the concept of orthogonal vectors defined as:
The term "orthogonal" in mathematics denotes a direction at a 90° angle. If two vectors u, and v form a right angle when they are perpendicular, or if the dot product they produce is zero, then they are said to be orthogonal.
The given vector is,
u = (5, −2, 8)
Let the vector orthogonal to this is,
v = (x, y, z)
Since vectors u and v are orthogonal,
Therefore,
u.v = 0
(5, −2, 8)(x, y, z) = 0
5x -2y + 8x = 0 .....(i)
To find the vectors choose x, y, and x such that equation one is satisfied.
So take x = 0, y = 4 and z = 1
Then (x, y, z) = (0, 4, 1) satisfy the equation (i)
Similarly, choose x = 8, y = 0, z = -5
Then (x, y, z) = (8, 0, -5) satisfy the equation (i)
Hence,
Two orthogonal vectors are (0, 4, 1) and (8, 0, -5)
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Two vectors orthogonal to u=(5, -2, 8) could be v=(2, 5, 0) and u'=(-2, -5, 0). Both vectors satisfy the definition of orthogonality, having a dot product with u equal to zero.
Explanation:To find two vectors that are orthogonal to the given vector u = (5, -2, 8), we need to find two vectors that have a dot product with vector u equal to zero. This is because orthogonality (perpendicularity in 3D space) is defined by a zero dot product.
For instance, let's calculate the dot product of u with v = (2, 5, 0). It'll be (5*2) + ((-2)*5) + (8*0) = 0. Therefore, vector v = (2, 5, 0) is orthogonal to u.
For a vector in the opposite direction, we simply need to multiply every component of vector v by -1. Consequently, a vector u′ = (-2, -5, 0) is in the opposite direction of v but still orthogonal to u. So, our answer is vectors v = (2, 5, 0) and u' = (-2, -5, 0).
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You would like to make a nutritious meal of eggs, mixed vegetables and brown rice. The meal should provide at least 35 g of carbohydrates, at least 30 g of protein, and no more than 45 g of fat. One serving of eggs contains 2 g of carbohydrates, 18 g of protein, and 12 g of fat. A serving of mixed vegetables contains 14 g of carbohydrates, 15 g of protein, and 8 g of fat. A serving of rice contains 40 g of carbohydrates, 6 g of protein, and 1 g of fat. A serving of eggs costs $3.75, a serving of mixed vegetables costs $3.50, and a serving of rice costs $2. It is possible to order a partial serving, e.g. 0.75 servings of rice. Formulate a linear optimization model that could be used to determine the number of servings of eggs, mixed vegetables, and rice for your meal that meets the nutrition requirements at minimal cost.
Answer:
Let G be the number of eggs in the meal
V be the number of servings of mixed vegetables in the meal
R be the number of servings of brown rice in the meal
Objective function = Minimize 3.75G + 3.50V + 2R
Constraints:
2G + 14V + 40R ≥ 35(Carbohydrates)
18G + 15V + 6R ≥ 30(Protein)
12G + 8V + R ≤ 45(Fat)
G, M, B ≥ 0
Do you tailgate the car in front of you? About 35% of all drivers will tailgate before passing, thinking they can make the car in front of them go faster. Suppose that you are driving a considerable distance on a two-lane highway and are passed by 12 vehicles.
(a) Let r be the number of vehicles that tailgate before passing. Make a histogram showing the probability distribution of r for r = 0 through r = 12.
(b) Compute the expected number of vehicles out of 12 that will tailgate. (Round your answer to two decimal places.)
vehicles
(c) Compute the standard deviation of this distribution. (Round your answer to two decimal places.)
vehicles
Answer:
(a) The histogram is shown below.
(b) E (X) = 4.2
(c) SD (X) = 2.73
Step-by-step explanation:
Let X = r = a driver will tailgate the car in front of him before passing.
The probability that a driver will tailgate the car in front of him before passing is, P (X) = p = 0.35.
The sample selected is of size n = 12.
The random variable X follows a Binomial distribution with parameters n = 12 and p = 0.35.
The probability function of a binomial random variable is:
[tex]P(X=x)={n\choose x}p^{x}(1-p)^{n-x}[/tex]
(a)
For X = 0 the probability is:
[tex]P(X=0)={12\choose 0}(0.35)^{0}(1-0.35)^{12-0}=0.006[/tex]
For X = 1 the probability is:
[tex]P(X=1)={12\choose 1}(0.35)^{1}(1-0.35)^{12-1}=0.037[/tex]
For X = 2 the probability is:
[tex]P(X=2)={12\choose 2}(0.35)^{2}(1-0.35)^{12-2}=0.109[/tex]
Similarly the remaining probabilities will be computed.
The probability distribution table is shown below.
The histogram is also shown below.
(b)
The expected value of a Binomial distribution is:
[tex]E(X)=np[/tex]
The expected number of vehicles out of 12 that will tailgate is:
[tex]E(X)=np=12\times0.35=4.2[/tex]
Thus, the expected number of vehicles out of 12 that will tailgate is 4.2.
(c)
The standard deviation of a Binomial distribution is:
[tex]SD(X)=np(1-p)[/tex]
The standard deviation of vehicles out of 12 that will tailgate is:
[tex]SD(X)=np(1-p)=12\times0.35\times(1-0.35)=2.73\\[/tex]
Thus, the standard deviation of vehicles out of 12 that will tailgate is 2.73.
To determine the probability distribution, create a histogram showing the possible values of r and their probabilities. The expected number of vehicles that will tailgate can be calculated by multiplying each value of r by its probability and summing up the results. The standard deviation can be found by calculating the variance and taking the square root of it.
Explanation:In order to determine the probability distribution of the number of vehicles that tailgate before passing, we need to consider the given information. We know that about 35% of all drivers tailgate. Since we are passed by 12 vehicles, the number of vehicles that tailgate can range from 0 to 12. To create a histogram showing the probability distribution, we need to calculate the probability of each possible value of r and represent them in a bar graph.
(b) To compute the expected number of vehicles that will tailgate, we need to multiply each possible value of r by its corresponding probability and sum up the results. This will give us the average number of vehicles that tailgate out of the 12 vehicles that passed us.
(c) The standard deviation of this distribution can be calculated by determining the variance and taking the square root of it. Variance is calculated by summing up the squared differences between each value of r and the expected value, multiplying each squared difference by its corresponding probability, and summing up the results.
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Exercise 1.28. We have an urn with m green balls and n yellow balls. Two balls are drawn at random. What is the probability that the two balls have the same color? (a) Assume that the balls are sampled without replacement. (b) Assume that the balls are sampled with replacement. (c) When is the answer to part (b) larger than the answer to part (a)? Justify your answer. Can you give an intuitive explanation for what the calculation tells you?
Answer:
Step-by-step explanation:
given that we have an urn with m green balls and n yellow balls. Two balls are drawn at random.
a) Assume that the balls are sampled without replacement.
m green and n yellow balls
For 2 balls to be drawn at the same colour
no of ways = either 2 green or 2 blue = mC2+nC2
Total no of ways = (m+n)C2
Prob =
= [tex]\frac{mC2 +nC2}{(m+n)C2} \\=\frac{m(m-1)+n(n-1)}{(m+n)(m+n-1)}[/tex]
=[tex]\frac{m^2+n^2-m-n}{(m+n)(m+n-1)}[/tex]
B) Assume that the balls are sampled with replacement
In this case, probability for any draw for yellow or green will be constant as
n/M+n or m/m+n respectively
Reqd prob = [tex](\frac{m}{m+n} )^2 +(\frac{n}{m+n} )^2[/tex]
=[tex]\frac{m^2+n^2}{(m+n)^2}[/tex]
c) Part B prob will be more than part a because with replacement prob is more than without replacement.
II time drawing same colour changes to m-1/.(m+n-1) if with replacement but same as m/(m+n) without replacement
[tex]\frac{m}{m+n} >\frac{m-1}{m+n-1} \\m^2+mn-m>m^2+mn-m-n\\n>0[/tex]
Since n>0 is true always, b is greater than a.
The question explores the concept of probability within scenarios of drawing balls of different colors from an urn, with and without replacement. It explores how the number of balls left in the urn changes the likelihood of drawing two balls of the same color. The answer is calculated using mathematical odds and conditions, showing that replacement affects probability especially when the total number of items (balls in this case) is small.
Explanation:The subject of this question is probability, specifically conditional probability and probability with and without replacement. Here are the calculations needed to answer the question:
(a) When the balls are drawn without replacement, the probability that the two balls drawn have the same color is the sum of the probability of drawing two green balls and the probability of drawing two yellow balls. The probability of drawing two green balls is (m/(m+n)) * ((m-1)/(m+n-1)). Similarly, the probability of drawing two yellow balls is (n/(m+n)) * ((n-1)/(m+n-1)). The sum of these two probabilities gives the required probability. (b) When the balls are drawn with replacement, the same logic applies; however, since the balls are replaced, the denominator term doesn't decrease for the second draw. Thus, the probability of drawing two green balls is (m/(m+n)) * (m/(m+n)), and the probability of two yellow balls is (n/(m+n)) * (n/(m+n)). (c) The answer to part (b) becomes larger than the answer to part (a) when m and n are small numbers. This is because, when m and n are small, the probability of drawing a similarly colored ball in the second draw becomes more significant if the ball is replaced after the first draw, compared to if it is not replaced, causing the probability with replacement to be higher.
In a nutshell, the calculation for probability tells us how likely an outcome is, given the mathematical odds and conditions (in this case, if the balls are replaced or not after drawing).
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The game of European roulette involves spinning a wheel with 37 slots: 18 red, 18 black, and 1 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. Gamblers can place bets on red or black. If the ball lands on their color, they double their money. If it lands on another color, they lose their money.(a) Suppose you play roulette and bet $3 on a single round. What is the expected value and standard deviation of your total winnings?(b) Suppose you bet $1 in three dierent rounds. What is the expected value and standard deviation of your total winnings?(c) How do your answers to parts (a) and (b) compare? What does this say about the riskiness of the two games?
Answer:
a) E(X) = -$0.0813 , s.d (X) = 3
b) E(X) = -$0.0813 , s.d (X) = 3
c) expected loss and higher stakes of loosing.
Step-by-step explanation:
Given:
- There are total 37 slots:
Red = 18
Black = 18
Green = 1
- Player on bets on either Red or black
- Wins double the bet money, loss the best is lost
Find:
a) Expected value of earnings X if we place a bet of $3
b) Expected value and standard deviation if we bet $1 each on three rounds
c) compare the two answers in a and b and comment on the riskiness of the two games
Solution:
- Define variable X as the total winnings per round. We will construct a distribution tables for total winnings per round for bets of $3 and $ 1:
- Bet: $3
X -3 3 E(X)
P(X) 1-0.48 = 0.5135 18/37 = 0.4864 3*(.4864-.52135) = -0.08
-The s.d(X) = sqrt(9*(0.5135 + 0.4864) - (-0.08)^2) = 3.0
- Bet: $1
X -1 1 E (X)
P(X) 1-0.48 = 0.5135 18/37 = 0.4864 1*(.4864-.5135) = -0.0271
-The s.d(X) = sqrt(1*(0.5135 + 0.4864) - (-0.0271)^2) = 0.999
- The expected value for 3 rounds is:
E(X_1 + X_2 + X_3) = E(X_1) + E(X_2) + E(X_3)
- The above X winnings are independent from each round, hence:
E(3*X_1) = 3*E(X_1) = 3*(-0.0271) = -0.0813
- The standard deviation for 3 rounds is:
sqrt(Var(X_1 + X_2 + X_3)) = sqrt(Var(X_1) + Var(X_2) + Var(X_3))
- The above X winnings are independent from each round, hence:
sqrt(Var(3*X_1)) = 3*Var(X_1) = 3*(0.999) = 2.9988
- For above two games are similar with an expected loss of $0.0813 for playing the game and stakes are very high due to high amount of deviation for +/- $3 of winnings.
In the game of European roulette, betting $1 in three different rounds is less risky than betting $3 in one round as it has a lower expected loss and lower variability.
Explanation:In the game of roulette, the probability of winning (i.e. the ball landing on either red or black) is 18/37, and the probability of losing (the ball landing on green) is 1/37.
(a) When you bet $3 on a single round, the expected value of your winnings is given by (18/37 * $6) - (19/37 * $3) = -$0.081, and the standard deviation is sqrt([$6-$(-0.081)]^2 * 18/37 + [-$3-$(-0.081)]^2 * 19/37) = $2.899.
(b) When you bet $1 in three different rounds, the expected value of your winnings is 3 * [ (18/37 * $2) - (19/37 * $1) ] = -$0.243, and the standard deviation is sqrt(3 * [(($2-$(-0.081))^2 * 18/37) + (($-1-$(-0.081))^2 * 19/37)]) = $1.578.
(c) If we compare the two games, it is clear that betting $1 in three different rounds decreases both the expected loss and the variability of the results, indicating that the latter game is less risky.
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A class receives a list of 20 study problems, from which 10 will be part of an upcoming exam. A student knows how to solve 15 of the problems. Find the probability that the student will be able to answer (a) all 10 questions on the exam, (b) exactly eight questions on the exam, and (c) at least nine questions on the exam.
Answer:
(a) 0.01625
(b) 0.3483
(c) 0.15170
Step-by-step explanation:
Given that there are total of 20 study problems from which 10 will come in the exam and A student knows how to solve 15 of the problems from those 20 total problems.
(a) To calculate the probability that the student will be able to answer all 10 questions on the exam, we know that:
students will be able answer all 10 questions in the exam only when all these 10 questions will be from the 15 problem which he know how to solve.
So the chances that he knows all 10 questions in the exam = [tex]^{15}C_1_0[/tex]
And total ways in which he answer 10 question from the 20 study problems =[tex]^{20}C_1_0[/tex]
Therefore, the Probability that the student will be able to answer all 10 questions on the exam = [tex]\frac{^{15}C_1_0}{^{20}C_1_0}[/tex] = [tex]\frac{15!}{5!\times 10!} \times \frac{10!\times 10!}{20!}[/tex] {Because [tex]^{n}C_r[/tex] = [tex]\frac{n!}{r!\times (n-r)!}[/tex] }
= 0.01625
(b) Probability that the student will be able to answer exactly eight questions on the exam = In the numerator there will be No. of ways that he answer exactly eight questions from the 15 problems he knows and the remaining 2 questions he solve from the 5 questions whose answer he doesn't know and in the denominator there will Total number of ways in which he answer 10 question from the 20 study problems.
So Required Probability = [tex]\frac{^{15}C_8 \times ^{5}C_2 }{^{20}C_1_0 }[/tex] = [tex]\frac{15!}{8!\times 7!}\times \frac{5!}{2!\times 3!}\times \frac{10!\times 10!}{20!}[/tex] = 0.3483
(c) To calculate the probability that student will be able to answer at least nine questions on the exam is given by that [He will be able to nine questions on the exam + He will be able to answer all 10 questions on the exam]
So no. of ways that he will be able to nine questions on the exam = He answer 9 questions from those 15 problems which he know and remaining one question from the other 5 questions we he don't know = [tex]^{15}C_9\times ^{5}C_1[/tex]
And no. of ways that he will be able to answer all 10 questions on the exam
= [tex]^{15}C_1_0[/tex]
So, the required probability = [tex]\frac{(^{15}C_9\times ^{5}C_1)+^{15}C_1_0}{^{20}C_1_0}[/tex] = 0.15170
The probability that the student will be able to answer all 10 questions on the exam is 0.01625.
How to calculate probability?The probability that the student will be able to answer all 10 questions on the exam will be:
= 15C10 / 20C10
= [15! / (5! × 10!)] × [10! × 20!/20!]
= 0.01625
The probability that the student will be able to answer exactly eight questions on the exam will be:
= (15C8 × 5C2) / 20C10
= 0.3483
The probability that the student will answer at least nine questions on the exam will be:
= [(15C9 × 5C1) + 15C10] / 20C10]
= 0.15170
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A certain standardized test's math scores have a bell-shaped distribution with a mean of 530 and a standard deviation of 119. Complete parts (a) through (c) (a) What percentage of standardized test scores is between 411 and 649? 68% (Round to one decimal place as needed.) (b) What percentage of standardized test scores is less than 411 or greater than 649? 1 32% (Round to one decimal place as needed.) (c) What percentage of standardized test scores is greater than 768? % (Round to one decimal place as needed.)
You can convert the given normal distribution to standard normal distribution and then use the z tables to find the needed probabilities.
Rounding to one places of decimal, we get the answers as:
[tex]P( 411 < X < 649 ) \approx 0.6826 = 68.2\%[/tex][tex]P(X < 411) + P(X > 649) \approx 0.3174 \approx 31.7\%[/tex][tex]P(X > 768) \approx 0.0228 \approx 2.3\%[/tex]How to get the z scores?If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.
If we have
[tex]X \sim N(\mu, \sigma)[/tex]
(X is following normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex])
then it can be converted to standard normal distribution as
[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]
Using the z scores will help to find the probabilities from the z tables(available online).
Let for the given test, the test scores be tracked by a random variable X, then by the given data, we have:
[tex]X \sim N(530, 119)[/tex]
The needed probabilities are
[tex]P( 411 < X < 649 ) = P(X < 649) - p(X < 411)\\[/tex][tex]P(X < 411) + P(X > 649) = 1 - P(411 \leq X \leq 649) = 1 - P(411 < X < 649)[/tex][tex]P(X > 768) = 1 - P(X \leq 768)[/tex]Converting the distribution to standard normal variate, the probabilities become
[tex]Z = \dfrac{X - 530}{119}\\\\Z \sim N(0, 1)[/tex]
The probabilities convert to
a) [tex]P( 411 < X < 649 ) = P(X < 649) - p(X < 411)\\[/tex]
[tex]P(\dfrac{411 - 530}{119} < Z < \dfrac{ 649 - 530}{119}) = P(-1 < Z < 1) = P(Z < 1) - P(Z < -1)[/tex]
(Know the fact that in continuous distribution, probability of a single point is 0, so we can write [tex]P(Z < a) = P(Z \leq a)[/tex] )
Also, know that if we look for Z = a in z tables, the p value we get is [tex]P(Z \leq a) = p \: value[/tex]
The p value at Z = 1 is 0.8413 and at Z = -1 is 0.1587,
Thus, [tex]P(411 < X < 649) = P(Z < 1) - P(Z < -1) = 0.8413 - 0.1587 = 0.6826[/tex]
b) [tex]P(X < 411) + P(X > 649) = 1 - P(411 \leq X \leq 649) = 1 - P(411 < X < 649)\\\\P(X < 411) + P(X > 649) = 1 - 0.6826 = 0.3174[/tex]
c) [tex]P(X > 768) = 1 - P(X \leq 768)[/tex]
[tex]P(X > 768) = 1 - P(Z < \dfrac{768 - 530}{119}) = 1 - P(Z < 2) = 1 - 0.9772 = 0.0228[/tex]
Rounding to one places of decimal, we get the answers as:
[tex]P( 411 < X < 649 ) \approx 0.6826 = 68.2\%[/tex][tex]P(X < 411) + P(X > 649) \approx 0.3174 \approx 31.7\%[/tex][tex]P(X > 768) \approx 0.0228 \approx 2.3\%[/tex]Learn more about standard normal distribution here:
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The dimensions of the bed of a dumptruck are 12.05 feet long, 3.86 feet wide and 5.11 feet 5.11 feet high, what is the volume of the dumptruck
Show your work below.
The volume of dump truck is 237.7 feet cubed.
Step-by-step explanation:
Given dimensions are;
Length of dump truck = 12.05 feet
Width of dump truck = 3.86 feet
Height of dump truck = 5.11 feet
Volume = Length * Width * Height
Volume = 12.05 * 3.86 * 5.11
Volume = 237.681 ft³
Rounding off to nearest tenth
Volume = 237.7 ft³
The volume of dump truck is 237.7 feet cubed.
Keywords: volume, multiplication
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A few fish species lay very big eggs and have big larvae with functioning gills from early on. However, most fish lay small eggs and the small larvae have no gills until later in life. In 2-3 sentences, discuss the need for gills in small vs. large fish larvae.
Answer:
For the exhange of ions and gases such as potassium, calcium and sodium, large fish larva have gills even in early stages. One other main reason is that large fish arvas mostly have circulating red cells after hatching phase while small fish larvae doesn't. So, for these reasons small fish larvae have no gills until later in life.
100pts: What is the remainder when 3^128 is divided by 17?
Answer:
The remainder is 9
Step-by-step explanation:
3^128 is divided by 17
find the value of 3^128 first.
3^128 = 1.17901845777E61
Then you divide by 17
1.17901845777E61 ÷ 17
= 6.93540269279E59
Approximately 6. 9340
6 remainder 9
Answer:
6 remander 1
Step-by-step explanation:
first you solve 3^128 which equals E61 then divide it by 17 which equals E59
Please help, thank you!
Two numbers total 63 and have a difference of 11. Find the two numbers.
Answer:
37 and 26
Step-by-step explanation:
Let the bigger number be x and the smaller number be y.
Then the two numbers total 63 will give us:
y+x=63------>eqn1
The two numbers have a difference of 11.
y-x=11------->eqn2
We add equations 1 and 2 to obtain:
2y=74
Divide both sides to get:
y=37
Put y=37 into eqn2 to get:
37-x=11
x=37-11
x=26
This larger number is 37 and the smaller number is 26
In humans, tongue rolling is a dominant trait, those with the recessive condition cannot roll their tongues. Dave can roll his tongue, but his father could not. He is married to Nancy, who can roll her tongue, but her mother could not. What is the probability that their first born child will be able to roll his tongue
Answer:
0.75 or 75%
Step-by-step explanation:
Let R be the dominant allele for rolling the tongue and r be the recessive allele for not rolling the tongue. If both Dave and Nancy can roll their tongues and had a parent that could not, they both have a heterozygous genotype (Rr).
The sample space for the genotype of their first born child is:
S={RR, Rr, rR, rr}
Only the homozygous recessive genotype rr makes it so that their child is not able to roll his tongue. Therefore, the probability that their first born child will be able to roll his tongue is:
[tex]P=\frac{3}{4}=0.75=75\%[/tex]
The manager of a radio station decides that on each successive evening (7 days per week), a Beethoven piano sonata will be played followed by a Beethoven symphony followed by a Beethoven piano concerto. For how many years could this policy be continued before exactly the same program would have to be repeated?
Answer:
3.945 years
Step-by-step explanation:
To answer this problem, one must know that Beethoven has composed 32 piano sonatas, 9 symphonies and 5 piano concertos.
The number of different arrangements that can be made by playing a sonata, then a symphony and then a piano concerto is:
[tex]n=32*9*5=1,440[/tex]
If a year has 365 days, the number of years that this daily policy could be continued before exactly the same program would have to be repeated is:
[tex]y=\frac{1,440}{365}=3.945\ years[/tex]
Suppose that P(A|B)=0.2, P(A|B')=0.3, and P(B)=0.7. What is the P(A)? Round your answer to two decimal places (e.g. 98.76).
Answer: [tex]P(A) = 0.23[/tex]
Step-by-step explanation:
Given :
[tex]P(A/B) = 0.2[/tex]
[tex]P(A/B^{1})=0.3[/tex]
[tex]P(B)= 0.7[/tex]
[tex]P(A) = ?[/tex]
From rules of probability :
[tex]P(A) = P(AnB) + P(A n B^{1})[/tex] ........................... equation *
[tex]P(A n B)[/tex] can be written as [tex]P(A/B)[/tex] x [tex]P(B)[/tex]
Also , [tex]P(A/B^{1})[/tex] can be written as [tex]P(A/B^{1})[/tex] x [tex]P(B^{1})[/tex]
substituting into equation * , we have
[tex]P(A) = P(A/B)[/tex][tex]P(B) + P(A/B^{1})P(B^{1})[/tex]
since P(B) = 0.7, then [tex]P(B^{1}) = 1 - P(B) = 0.3[/tex]
so , substituting each values , we have
[tex]P(A) = (0.2)(0.7) + (0.3)(0.3)[/tex]
[tex]P(A) = 0.14 + 0.09[/tex]
[tex]P(A) = 0.23[/tex]
Suppose that the national average for the math portion of the College Board's SAT is 515. The College Board periodically rescales the test scores such that the standard deviation is approximately 100. Answer the following questions using a bell-shaped distribution and the empirical rule for the math test scores.
(a) What percentage of students have an SAT math score greater than 615?
(b) What percentage of students have an SAT math score greater than 715?
(c) What percentage of students have an SAT math score between 415 and 515?
(d) What is the z-score for student with an SAT math score of 620?
(e) What is the z-score for a student with an SAT math score of 405?
Answer:
a) 16% of students have an SAT math score greater than 615.
b) 2.5% of students have an SAT math score greater than 715.
c) 34% of students have an SAT math score between 415 and 515.
d) [tex]Z = 1.05[/tex]
e) [tex]Z = -1.10[/tex]
Step-by-step explanation:
To solve this question, we have to understand the normal probability distribution and the empirical rule.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Empirical rule
The Empirical Rule states that, for a normally distributed random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we have that:
[tex]\mu = 515, \sigma = 100[/tex]
(a) What percentage of students have an SAT math score greater than 615?
615 is one standard deviation above the mean.
68% of the measures are within 1 standard deviation of the mean. The other 32% are more than 1 standard deviation from the mean. The normal probability distribution is symmetric. So of those 32%, 16% are more than 1 standard deviation above the mean and 16% more then 1 standard deviation below the mean.
So, 16% of students have an SAT math score greater than 615.
(b) What percentage of students have an SAT math score greater than 715?
715 is two standard deviations above the mean.
95% of the measures are within 2 standard deviations of the mean. The other 5% are more than 2 standard deviations from the mean. The normal probability distribution is symmetric. So of those 5%, 2.5% are more than 2 standard deviations above the mean and 2.5% more then 2 standard deviations below the mean.
So, 2.5% of students have an SAT math score greater than 715.
(c) What percentage of students have an SAT math score between 415 and 515?
415 is one standard deviation below the mean.
515 is the mean
68% of the measures are within 1 standard deviation of the mean. The normal probability distribution is symmetric, which means that of these 68%, 34% are within 1 standard deviation below the mean and the mean, and 34% are within the mean and 1 standard deviation above the mean.
So, 34% of students have an SAT math score between 415 and 515.
(d) What is the z-score for student with an SAT math score of 620?
We have that:
[tex]\mu = 515, \sigma = 100[/tex]
This is Z when X = 620. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{620 - 515}{100}[/tex]
[tex]Z = 1.05[/tex]
(e) What is the z-score for a student with an SAT math score of 405?
We have that:
[tex]\mu = 515, \sigma = 100[/tex]
This is Z when X = 405. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{405 - 515}{100}[/tex]
[tex]Z = -1.10[/tex]
a. Approximately 15.87% of students have an SAT math score greater than 615.
b. Approximately 2.28% of students have an SAT math score greater than 715.
c. 68% percentage of students have an SAT math score between 415 and 515
d. 1.05 is the z-score for student with an SAT math score of 620
e. -1.1 is the z-score for a student with an SAT math score of 405
To answer these questions, we can use the properties of a bell-shaped distribution and the empirical rule. The empirical rule states that for a bell-shaped distribution:Approximately 68% of the data falls within one standard deviation of the mean. Approximately 95% of the data falls within two standard deviations of the mean.Approximately 99.7% of the data falls within three standard deviations of the mean.
Given information: Mean (μ) = 515 and Standard Deviation (σ) = 100
(a) To find this, we need to calculate the z-score for a score of 615 and then find the percentage of data above that z-score using the standard normal distribution table.
z-score = (X - μ) / σ
z-score = (615 - 515) / 100
z-score = 1
Using the standard normal distribution table (or calculator), we find that approximately 84.13% of the data is below a z-score of 1. Since the distribution is symmetric, the percentage above the z-score of 1 is also approximately 100% - 84.13% = 15.87%.
Therefore, approximately 15.87% of students have an SAT math score greater than 615.
(b) We repeat the same process for a score of 715.
z-score = (715 - 515) / 100
z-score = 2
Using the standard normal distribution table (or calculator), we find that approximately 97.72% of the data is below a z-score of 2. The percentage above the z-score of 2 is approximately 100% - 97.72% = 2.28%.
Therefore, approximately 2.28% of students have an SAT math score greater than 715.
(c) We can use the empirical rule to find the percentage of students within one standard deviation of the mean and then subtract that from 100% to find the percentage between 415 and 515.
Percentage between 415 and 515 ≈ 68%
(d) We can calculate the z-score as follows:
z-score = (620 - 515) / 100
z-score = 1.05
(e) We can calculate the z-score as follows:
z-score = (405 - 515) / 100
z-score = -1.1
Remember that a negative z-score indicates a value below the mean.
Note: These calculations assume a normal distribution, and the actual percentages may vary slightly due to the discrete nature of test scores and rounding in calculations.
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Suppose there are two neighbors, Jared and Paul. These two neighbors have the same size lawn and the same amount of hedges. Each week the two neighbors mow their own lawns and trim their own hedges. It takes Jared 30 minutes to mow his lawn and 60 minutes to trim his hedges for a total time of 90 minutes of yard work. It takes Paul 120 minutes to mow his lawn and 90 minutes to trim his hedges, for a total of 210 minutes of yard work. Could the two neighbors gain by specializing and trading lawn services?, If so where should they specialize and how much time could the save each week?.
Answer:
Paul should trim both hedges and Jared should mow both lawns.
Each neighbor would save 30 minutes per week
Step-by-step explanation:
The time each neighbor takes to finish each task is presented below:
[tex]\begin{array}{ccc}&Paul&Jared\\Mow&120&30\\Trim&90&60\end{array}[/tex]
Jared is better at mowing the lawn than trimming hedges, while Paul is better at trimming hedges than mowing the lawn. Therefore, the two neighbors could gain by specializing and trading lawn services if Paul were to trim both hedges and Jared were to mow the lawns.
The time saved by each is:
[tex]P = 210 -90-90=30\ min\\J = 90-30-30=30\ min[/tex]
Defects in a product occur at random according to a Poisson distribution with parameter ???? = 0.04. What is the probability that a product has one or more defects? If the manufacturing process of the product is improved and the occurrence rate of defects is cut in half to ???? = 0.02. What effect does this have on the probability that the product has one or more defects?
Answer:
When the occurrence rate of defect is cut in half, the probability of a product having one or more defects drops drastically (0.0392 to 0.0198), to almost the half of its original value too.
Step-by-step explanation:
Poisson distribution formula
P(X=x) = f(x) = (λˣe^(-λ))/x!
λ = 0.04.
And the probability that a products one or more defects is the same thing as 1 minus the probability that a product has no defect.
P(X ≥ 1) = 1 - P(X = 0) = 1 - f(0)
P(X ≥ 1) = 1 - (0.04⁰e^(-0.04))/0! = 1 - 0.9608 = 0.0392
When the occurrence rate of defect is cut in half, that is, λ = 0.02,
P(X ≥ 1) = 1 - P(X = 0) = 1 - f(0)
P(X ≥ 1) = 1 - (0.02⁰e^(-0.02))/0! = 1 - 0.9802 = 0.0198
When the occurrence rate of defect is cut in half, the probability of a product having one or more defects drops drastically, to almost the half of its original value too.
Hope this helps!
The distribution of the average amount of sleep per night gotten by college students is roughly bell-shaped with mean 412 minutes and standard deviation 68 minutes. The proportion of those who get an average of less than eight hours (480 minutes) of sleep per night is about:
Answer:
[tex]P(X<480)=P(\frac{X-\mu}{\sigma}<\frac{480-\mu}{\sigma})=P(Z<\frac{480-412}{68})=P(z<1)[/tex]
[tex]P(z<1)=0.841[/tex]
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the scores of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(412,68)[/tex]
Where [tex]\mu=412[/tex] and [tex]\sigma=68[/tex]
We are interested on this probability
[tex]P(X<480)[/tex]
And the best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(X<480)=P(\frac{X-\mu}{\sigma}<\frac{480-\mu}{\sigma})=P(Z<\frac{480-412}{68})=P(z<1)[/tex]
And we can find this probability using the normal standard table or excel and we got:
[tex]P(z<1)=0.841[/tex]
Jack and Rodger both produce Sandwiches and Pies, and they both have 300 minutes of time available. It takes Jack 1 minutes to make a Sandwich, and 7 minutes to make a Pie. It takes Rodger 7 minutes to make a Sandwich and 1 minutes to make a Pie. What is the largest number of Sandwiches that Jack would be willing to trade away to get 4 Pies from Rodger
Answer:
28 sandwiches
Step-by-step explanation:
If Jack takes 7 minutes to make a pie, the time that would take Jack to produce 4 pies is:
[tex]t=4*7=28\ minutes[/tex]
Jack would be willing to trade away the amount of sandwiches he is able to produce in 28 minutes to get 4 pies from Rodger. In 28 minutes, the number of sandwiches Jack can produce is:
[tex]S=1*28=28\ sandwiches[/tex]
Jack would be willing to trade away 28 sandwiches for 4 pies.