A consumer organization inspecting new cars found that many had appearance defects. While none had more than 3 defects, 7% had three, 11% had two, and 21% had one defect. Find the expected number of appearance defects in a new car, and the standard deviation.

[tex]A) -27a^3 b^6 + 8a^9 b^{12}\\D) 27a^3 b^6 + 8a^9 b^{12}[/tex]

Step-by-step explanation:

Here, the given expressions are:

[tex]A) -27a^3 b^6 + 8a^9 b^{12}\\= (-3)^3(a^3)(b^2)^3 + (2)^3(a^3)3(b^4)^3\\= (-3ab^2)^3 +(2a^3b^4)^3[/tex]

So, the above expression is "sum of cubes".

[tex]B) -9a^3 b^6 + a^9 b^{10}\\[/tex]

But (-9) can not be expressed as a Perfect cube root.

So, the above expression is not "sum of cubes".

[tex]C) 9a^3 b^6 + 8a^9 b^{12}\\[/tex]

But (9) can not be expressed as a Perfect cube root.

So, the above expression is not "sum of cubes".

[tex]D) 27a^3 b^6 + 8a^9 b^{12}\\\\= (3)^3a^3(b^2)^3 + (2)^3(a^3)^3(b^4)^3\\= (3ab^2)^3+ (2a^3b^4)^3[/tex]

So, the above expression is  "sum of cubes".

23) x = [tex]\frac{-60}{9}[/tex] = -6.666.

24) x = [tex]\frac{-12}{7}[/tex] = -1.7142.

25) x = [tex]\frac{-37}{5}[/tex] = -7.4.

Step-by-step explanation:

Step 1; For [tex]\frac{x+6}{3}[/tex] = [tex]\frac{x+4}{12}[/tex], we cross multiply the denominators and get,

3 × (x + 4) = 12 × (x + 6),

3x + 12 = 12x + 72.

We take all the x terms to the LHS and keep the constants on the RHS.

3x - 12x = 72 - 12,

-9x = 60, x = [tex]\frac{-60}{9}[/tex] = -6.6666.

Step 2; For [tex]\frac{-5}{x-4}[/tex] = [tex]\frac{9}{x+12}[/tex], we cross multiply the denominators and get,

-5 × (x + 12) = 9 × (x - 4),

-5x - 60 = 9x - 36.

We take all the x terms to the LHS and keep the constants on the RHS.

-5x - 9x = -36 + 60,

-14x = 24, x = [tex]\frac{-24}{14}[/tex] = -1.7142.

Step 3; For [tex]\frac{6}{11}[/tex] = [tex]\frac{x-1}{x-8}[/tex], we cross multiply the denominators and get,

6 × (x - 8) = 11 × (x - 1),

6x - 48 = 11x - 11.

We take all the x terms to the LHS and keep the constants on the RHS.

6x - 11x = -11 + 48,

-5x = 37, x = [tex]\frac{-37}{5}[/tex] = -7.4.

Answer:

Pressure would be approximately 4.66 pounds.

Step-by-step explanation:

Given:

Volume of gas (V) = 25 cubic cm

Pressure of the  gas (P) = 11 pounds

We need to find the pressure when volume is 59 cubic cm.

Solution:

Now Given:

[tex]V[/tex] ∝ [tex]\frac{1}{P}[/tex]

so we can say that;

[tex]V =\frac kP[/tex]

where k is a constant.

When V = 25 cubic cm, P =11 pounds.

[tex]25 = \frac{k}{11}\\\\k= 25\times 11 = 275\ cm^3.pounds[/tex]

So the equation becomes as.

[tex]V = \frac{275}{P}[/tex]

Now we need to find the pressure when Volume is 59 cubic cm.

[tex]59 =\frac{275}{P}\\\\P=\frac{275}{59}\\\\P\approx 4.66\ pounds[/tex]

Hence Pressure would be approximately 4.66 pounds.

Answer:

y = 3.6(sine( 6.2(x-4.2))+4.4

Step-by-step explanation:

(8.2-.6)/2 = altitude = 3.6

6.2 = Wavelength

(8.2+.6)/2 = 4.4 The "line" (the horizontal central line thingy whose name I forgot cuz it's 12:00)

4.2 = x shift

y = 3.6(sine( 6.2(x-4.2))+4.4

Answer:

The answer to your question is 2² 3² or (4)(9)

Step-by-step explanation:

Data

factor 36

Process

1.- Divide 36 by prime numbers starting from 2, then 3, 5, 7, etc.

                          36  2

                           18  2

                            9   3

                            3   3

                             1

2.- Write 36 as a composition of prime numbers

                         36 = 2²3²

3.- The prime factors of 36 are 2² x 3²

Answer:

Fruit smoothie: $3.4

Protein bar: $2.5

Step-by-step explanation:

Let x represent cost of fruit smoothie and y represent cost of protein bar.

We have been given that Jake buys a fruit smoothie and a protein bar for​ $5.90. We can represent this information in an equation as:

[tex]x+y=5.90...(1)[/tex]

[tex]x=5.90-y...(1)[/tex]

We are also told that Kobe buys 2 fruit smoothies and 4 protein bars. He pays​ $16.80. We can represent this information in an equation as:

[tex]2x+4y=16.80...(2)[/tex]

Upon substituting equation (1) in equation (2), we will get:

[tex]2(5.90-y)+4y=16.80[/tex]

[tex]11.80-2y+4y=16.80[/tex]

[tex]2y=16.80-11.80[/tex]

[tex]2y=5[/tex]

[tex]y=\frac{5}{2}=2.5[/tex]

Therefore, each protein bar costs $2.5.

Upon substituting [tex]y=2.5[/tex] in equation (1), we will get:

[tex]x=5.90-2.5=3.4[/tex]

Therefore, each fruit smoothie costs $3.4.

Each fruit smoothie costs [tex]3.40\ dollars[/tex], and each protein bar costs [tex]2.50\ dollars[/tex].

To solve for the cost of each fruit smoothie [tex](\( x \))[/tex] and each protein bar [tex](\( y \))[/tex], we'll use the given system of equations:

1. [tex]\( x + y = 5.90 \)[/tex]

2. [tex]\( 2x + 4y = 16.80 \)[/tex]

Let's solve this step by step.

Step 1: Solve the first equation for [tex]\( x \)[/tex]

[tex]\[ x + y = 5.90 \][/tex]

[tex]\[ x = 5.90 - y \][/tex]

Step 2: Substitute [tex]\( x = 5.90 - y \)[/tex] into the second equation:

[tex]\[ 2(5.90 - y) + 4y = 16.80 \][/tex]

[tex]\[ 11.80 - 2y + 4y = 16.80 \][/tex]

[tex]\[ 2y = 16.80 - 11.80 \][/tex]

[tex]\[ 2y = 5 \][/tex]

[tex]\[ y = \frac{5}{2} \][/tex]

[tex]\[ y = 2.50 \][/tex]

Step 3: Substitute [tex]\( y = 2.50 \)[/tex] back into [tex]\( x = 5.90 - y \)[/tex]

[tex]\[ x = 5.90 - 2.50 \][/tex]

[tex]\[ x = 3.40 \][/tex]

We are given a velocity equation, and from that, we want to find the initial velocity such that we know the velocity for a given time.

We will see that the initial velocity is 124 ft/s

-------------------------------

Let's see how to solve this:

We have that the velocity equation:

v(t) = v₀ - (32 ft/s^2)*t

Where I added the units of the gravitational acceleration, which are in ft over seconds squared.

We want to get the value of the initial velocity, v₀, given that after 3 seconds the velocity is 28ft/s.

This means that:

v(3s) = 28 ft/s = v₀ - (32 ft/s^2)*3s

We can solve this for v₀:

28 ft/s = v₀ - (32 ft/s^2)*3s

28 ft/s + (32 ft/s^2)*3s = v₀

124 ft/s = v₀

So we can see that the initial velocity is 124 ft/s

If you want to learn more, you can read:

https://brainly.com/question/9163788

Answer:

164 pounds

Step-by-step explanation:

Please see attached picture for full solution.

Answer:

Step-by-step explanation:

What is the question

The solution for the equation is [tex]p=\frac{9}{8}[/tex]

Explanation:

The given equation is [tex]p-\frac{1}{4}=\frac{7}{8}[/tex]

We need to solve the equation.

The solution of the equation can be determined by finding the value for p.

Thus, from the equation, let us add both sides of the equation by [tex]\frac{1}{4}[/tex]

Hence, we have,

[tex]p-\frac{1}{4}+\frac{1}{4}=\frac{7}{8}+\frac{1}{4}[/tex]

Simplifying the equation, we get,

[tex]p=\frac{7}{8}+\frac{1}{4}[/tex]

Taking LCM for 4 and 8, we get,

[tex]p=\frac{7+2}{8}[/tex]

Adding the numerator, we have,

[tex]p=\frac{9}{8}[/tex]

Thus, the value of p is [tex]p=\frac{9}{8}[/tex]

Hence, the solution for the equation is [tex]p=\frac{9}{8}[/tex]

Assuming it is .005y^2 + 10y not .005*y*2 + 10y

Profit = Revenue - Cost

Profit = (.005y^2 + 10y) - (20y + 1,000,000)

Profit at 30,000 cars so y = 30000

Profit = (.005(30000)^2 + 10(30000)) - (20(30000) + 1,000,000)

Profit = $3,200,000

Answer:

[tex]TA=(144+36\sqrt{3})\ units^2[/tex]

Step-by-step explanation:

we know that

The total area or surface area of the regular pyramid is equal to the area of the triangular base plus the area of its three lateral triangular faces

so

step 1

Find the area of the triangular base B

Is an equilateral triangle

Applying the law of sines

[tex]B=\frac{1}{2}(12^2)sin(60^o)[/tex]

[tex]B=\frac{1}{2}(144)\frac{\sqrt{3}}{2}[/tex]

[tex]B=36\sqrt{3}\ units^2[/tex]

step 2

Find the area of the lateral triangular faces

[tex]A=3[\frac{1}{2}(12)h][/tex]

Find the height

Applying the Pythagorean Theorem

[tex]10^2=6^2+h^2[/tex]

[tex]h^2=100-36\\h^2=64\\h=8\ units[/tex]

Find the area of the lateral triangular faces

[tex]A=3[\frac{1}{2}(12)8]=144\ units^2[/tex]

therefore

The total area is

[tex]TA=(144+36\sqrt{3})\ units^2[/tex]

Answer:

P ( E_1*E_2*E_3*E_4 ) = 0.1055

Step-by-step explanation:

Given:

- 52 cards are dealt in 1 , 2 , 3 , 4 hands.

- Events:

             E_1       Hand 1 has exactly 1 ace

             E_2       Hand 2 has exactly 1 ace

             E_3       Hand 3 has exactly 1 ace

             E_4       Hand 4 has exactly 1 ace

Find:

p =P ( E_1*E_2*E_3*E_4 )

Solution:

Multiplication rule.

- For n ε N and events E_1 , E_2 , ... , E_n:

P ( E_1*E_2*......*E_n ) = P (E_1)*P(E_2|E_1)*P(E_3|E_2*E_1)*......*(E_n|E_1*E_2...E_n-1 )

- So for these events calculate 4 probabilities:-

-  For E_1, is to choose an ace multiplied by the number of ways to choose remaining 12 cards out of 48 non-aces:

                               P ( E_1 ) = 4C1 * 48C12 / 52C13

- For E_2 | E_1 , one ace and 12 other cards have already been chosen. there are 39C13 equally likely hands. The number of different one ace hand 2 is the number of ways to choose an ace from 3 remaining multiplied by the number of ways to choose the remaining 12 from 36, we have:

                               P ( E_2 | E_1  ) = 3C1 * 36C12 / 39C13

                               P ( E_3| E_2*E_1  ) = 2C1 * 24C12 / 26C13

                               P ( E_4 | E_3*E_2*E_1  ) = 1C1*12C12 / 13C13 = 1

- So the multiplication rule for n = 4 is as follows:

     P ( E_1*E_2*E_3*E_4 ) = P (E_1)*P(E_2|E_1)*P(E_3|E_2*E_1)*P ( E_4 | E_3*E_2*E_1  ) = [ 4C1 * 48C12 / 52C13 ] * [ 3C1 * 36C12 / 39C13 ] * [ 2C1 * 24C12 / 26C13 ]

     P ( E_1*E_2*E_3*E_4 ) = [ 4!*48! / (12!)^4 ] / [ 52! / (13!)^4 ]

     P ( E_1*E_2*E_3*E_4 ) = [ 4!*13^4 / (52*51*50*49) ]

    P ( E_1*E_2*E_3*E_4 ) = 0.1055

The probability that each hand in a deck of 52 cards gets exactly one ace is approximately 10.5%.

To determine the probability that each hand in a randomly divided deck of 52 cards has exactly one ace, we use the concept of conditional probability.
Let's find it step by step

Step 1 : consider the event E1 that the first hand has exactly one ace:

There are 4 aces and 52 total cards. The probabilities for drawing an ace for the first hand are affected by the decreasing number of both aces and cards.

The probability of the first hand receiving one ace is calculated as:

P(E1) = (4/52) * (48/51) * (47/50) * ... * (36/39)

Step 2 : consider the event E2 that the second hand receives exactly one ace, given that the first hand already has one:

With one ace already given to the first hand, there are 3 aces remaining and 39 cards left for the second hand.

The probability is calculated as:

P(E2|E1) = (3/39) * (35/38) * ... * (25/26)

Step 3 : Proceed similarly for the third and fourth hands:

P(E3|E1E2) = (2/26) * ... * (12/13)

P(E4|E1E2E3) = 1 (since only one ace remains for the last hand)

Step 4 : Using the multiplication rule, the overall probability P(E1E2E3E4) is calculated by multiplying the individual probabilities:

P(E1E2E3E4) = P(E1) * P(E2|E1) * P(E3|E1E2) * P(E4|E1E2E3)

Step 5 : After performing the calculations, we find:

The combined probability P(E1E2E3E4) = (4/52)*(3/39)*(2/26)(1/13) after simplifying is approximately 0.105 or 10.5%.

Answer: 2.1 feet

Step-by-step explanation:

The ladder forms a right angle triangle with the wall and the ground. The length of the ladder represents the hypotenuse of the right angle triangle. The height from the top of the ladder to the base of the wall represents the opposite side of the right angle triangle.

The distance, d from the bottom of the ladder to the base of the wall represents the adjacent side of the right angle triangle.

To determine the distance, d from the bottom of the ladder to the base of the wall, we would apply we would apply the cosine trigonometric ratio.

Cos θ = adjacent side/hypotenuse. Therefore,

Cos 65 = d/5

d = 5Cos 65 = 5 × 0.4226

d = 2.1 feet

Answer:

  -5

Step-by-step explanation:

The power rule can be used.

  f(x) = 5x^-1

  f'(x) = 5(-1)x^(-1-1)

  f'(x) = -5x^-2

Then ...

  f'(-1) = -5(-1)^-2

  f'(-1) = -5

_____

The attached graph shows the value of the derivative at x=-1, along with a tangent line having that slope at the point (-1, f(-1)).

Answer:It depends

Step-by-step explanation:

Answer:

Step-by-step explanation:

We would apply the formula for

exponential decay which is expressed as

A = P(1 - r)^t

Where

A represents the population after t years.

t represents the number of years.

P represents the initial population.

r represents rate of growth.

From the information given,

P = 1,450,000

r = 0.99% = 0.99/100 = 0.0099

t = x years

Therefore, an exponential equation to represent this situation after x years is

A = 1450000(1 - 0.0099)^t

A = 1450000(0.9901)^t

Answer: $406

Step-by-step explanation:

Answer: she earned $406 last week.

Step-by-step explanation:

Last week, she worked 2 hours on Monday, 10 hours on Tuesday, and 9 hours on Wednesday. This means that the number of hours that she worked for the first three days is

2 + 10 + 9 = 21 hours

She had Thursday off, and then she worked 8 hours on Friday. Therefore, the total number of hours that she worked for the week is 21 + 8 = 29 hours.

If Nanette earns $14 per hour, then the total amount of money that Nanette earned in all last week is

29 × 14 = $406

This is a very simple and easy problem. I'm not sure why you need someone else to solve it, but I hope this helps

a. Linear equation:

Let x be amount of movies rented

$8 + ($2.50 * x)

b.

$8 + ($2.50 * 10)

= $8 + $25.0

= $33

The probability of drawing two chips of different colors from the bag is 35/33.

The probability of drawing the chips:

Calculate the total number of ways to draw 2 chips: 12 chips total, so 12C2 = 66 ways.

Calculate the number of ways to draw 2 chips of different colors: (5 blue chips × 7 non-blue chips) + (7 non-blue chips × 5 blue chips) = 70 ways.

Probability = Number of favorable outcomes / Total outcomes = 70/66 = 35/33.

the probability that the two selected chips are of different colors is [tex]\( \frac{94}{144} \), which simplifies to \( \frac{47}{72} \).[/tex]

To find the probability that the two selected chips are of different colors, we can use the concept of complementary probability.

The complementary event of selecting two chips of different colors is selecting two chips of the same color.

Let's calculate the probability of selecting two chips of the same color and then subtract that from 1 to find the probability of selecting two chips of different colors.

1. Probability of selecting two blue chips:

[tex]\[ P(\text{blue, blue}) = \frac{5}{12} \times \frac{5}{12} = \frac{25}{144} \][/tex]

2. Probability of selecting two red chips:

[tex]\[ P(\text{red, red}) = \frac{4}{12} \times \frac{4}{12} = \frac{16}{144} \][/tex]

3. Probability of selecting two yellow chips:

[tex]\[ P(\text{yellow, yellow}) = \frac{3}{12} \times \frac{3}{12} = \frac{9}{144} \][/tex]

Now, let's add these probabilities together because any of these scenarios results in two chips of the same color:

[tex]\[ P(\text{same color}) = P(\text{blue, blue}) + P(\text{red, red}) + P(\text{yellow, yellow}) \]\[ P(\text{same color}) = \frac{25}{144} + \frac{16}{144} + \frac{9}{144} = \frac{50}{144} \][/tex]

Finally, we subtract this probability from 1 to find the probability of selecting two chips of different colors:

[tex]\[ P(\text{different colors}) = 1 - P(\text{same color}) \]\[ P(\text{different colors}) = 1 - \frac{50}{144} = \frac{144}{144} - \frac{50}{144} = \frac{94}{144} \][/tex]

So, the probability that the two selected chips are of different colors is [tex]\( \frac{94}{144} \), which simplifies to \( \frac{47}{72} \).[/tex]

Answer:

It is  [tex]11\frac{9}{10}[/tex]  miles far from Chester to Durbin.

Step-by-step explanation:

Given:

Its 10 3/5 miles from Alston to Barton and 12 1/2 miles from Barton to Chester. The distance from Alston to Durbin, via barton and Chester, is 35 miles.

Now, to find the distance from Chester to durbin.

Distance from Alston to Barton = [tex]10\frac{3}{5} =\frac{53}{5} \ miles.[/tex]

Distance from Barton to Chester = [tex]12\frac{1}{2}\ miles =\frac{25}{2} \ miles.[/tex]

As, given the distance from Alston to Durbin, via barton and Chester, is 35 miles.

Thus, the total distance = 35 miles.

So, we add the distance of Alston to Barton and Barton to Chester and get the distance from Alston to Chester:

[tex]\frac{53}{5} +\frac{25}{2}[/tex]

[tex]=\frac{106+125}{10}[/tex]

[tex]=\frac{231}{10} \ miles.[/tex]

Distance from Alston to Chester  [tex]=\frac{231}{10} \ miles.[/tex]

Now, to get the distance from Chester to durbin we subtract distance from Alston to Chester from the total distance:

[tex]35-\frac{231}{10} \\\\=\frac{350-231}{10} \\\\=\frac{119}{10} \\\\=11\frac{9}{10}\ miles.[/tex]

Therefore, it is  [tex]11\frac{9}{10}[/tex]  miles far from Chester to Durbin.

Answer:

The range of values for the sample mean is between a lower limit of 19 and an upper limit of 21.

Step-by-step explanation:

sample mean = 20

sd = 10

n = 25

standard error = 1

Lower limit of sample mean = sample mean - standard error = 20 - 1 = 19

Upper limit of sample mean = sample mean + standard error = 20 + 1 = 21

The range of values for the sample mean is between 19 and 21.

Answer:

12

Step-by-step explanation:

15000x = 9000x +72000

6000x = 72000

x = 12

Answer: the number of units that must be sold to break even is 12

Step-by-step explanation:

The cost function is expressed as

C(x)= 9000x +72000

The revenue function is expressed as

R(x) = 15000x

Profit = Revenue - cost

At the point of break even, the total revenue is equal to the total cost. This means that profit is zero. The expression becomes

Revenue - cost = 0

Revenue = cost

R(x) = C(x)

Therefore,

15000x = 9000x +72000

15000x - 9000x = 72000

6000x = 72000

x = 72000/6000

x = 12

Answer:3

Step-by-step explanation:

Answer:

3

Step-by-step explanation:

khan acadamy hope this helps

Answer:

Total number of possibilities 2,095,681,645,538.

Step-by-step explanation:

The variables can be 1 to 8 characters long.

The first space can be filled by any of the 26 letters.

The remaining n places can be filled by any of the 26 letters or any of the 10 digits.

For a single character variable the number of ways to select a variable name is,

n (1 character) = 26

For two character variable the number of ways to select a variable name is,

n (2 character) = 26 × 36 = 936

For three character variable the number of ways to select a variable name is,

n (3 character) = 26 × 36 × 36 = 26 × 36² = 33,696

For four character variable the number of ways to select a variable name is,

n (4 character) = 26 × 36 × 36 × 36 = 26 × 36³ = 1,213,056

And so on.

Similarly for the eight character variable the number of ways to select a variable name is,

n (8 character) = 26 × 36 × 36... × 36 = 26 × 36⁷ = 2,037,468,266,496

Total number of possibilities 2,095,681,645,538.

To solve the equation sin 2x - sin 4x = 0, we apply the identity for the difference of two sines and set each term equal to zero. The solutions in the interval [0, 2π) are x = 0, π/6, 5π/6, π.

The equation given is sin 2x - sin 4x = 0. To find the solutions to this equation in the interval [0, 2π), we can use the trigonometric identity for the difference of two sines, sin A - sin B = 2 sin((A - B)/2) cos((A + B)/2). Applying this identity:

2 sin(-2x/2) cos(6x/2) = 0
2 sin(-x) cos(3x) = 0

Since sin(-x) = -sin(x), we can rewrite the equation further:

-2 sin(x) cos(3x) = 0

To find the solutions, set each part equal to zero:

sin(x) = 0

cos(3x) = 0

For sin(x) = 0, the solutions in [0, 2π) are x = 0, π, 2π. However, since the interval is [0, 2π), 2π is not included.

For cos(3x) = 0, the solutions are x = π/6, 5π/6 since cos(x) has a period of 2π and 3x adds additional repetitions of the solutions in the interval.

The complete set of solutions in the interval [0, 2π) are therefore:

0

π/6

5π/6

π

The inequality is:

[tex]n > \frac{m}{4}[/tex]

Membership of the pool will be less expensive until number of visits to the pool is one fourth of the membership amount

Solution:

Given that,

A pool charges $4 each visit or you can buy a membership

Let "n" be the number of times you visit the pool

Let the membership amount of the pool be "m"

A pool charges $4 each visit

Therefore, cost for "n" visit is: $ 4n

The inequality showing that a membership is less expensive than paying each visit to the pool is:

4n > m

Divide both sides by "4"

[tex]n > \frac{m}{4}[/tex]

Therefore, membership of the pool will be less expensive until number of visits to the pool is one fourth of the membership amount

Answer:

[tex]\text{Area of margin}=145\text{ cm}^2[/tex]

Step-by-step explanation:

We have been given that a painting is drawn on a cardboard 22 cm long and 12 cm wide such that there is a margin of 2.5 meter cm along each side. We are asked to find the area of the margin.

The total area of the margin would be equal to area of whole cardboard minus area of painting.

[tex]\text{Area of whole cardboard}=22\text{ cm}\times 12\text{ cm}[/tex]

[tex]\text{Area of whole cardboard}=264\text{ cm}^2[/tex]

Since there is a margin of 2.5 meter cm along each side, so sides of painting would be 2,5 cm smaller on four sides. The sides painting would be [tex]22-5=17[/tex] and [tex]12-5=7[/tex].

[tex]\text{Area of painting}=17\text{ cm}\times 7\text{ cm}[/tex]

[tex]\text{Area of painting}=119\text{ cm}^2[/tex]

[tex]\text{Area of margin}=264\text{ cm}^2-119\text{ cm}^2[/tex]

[tex]\text{Area of margin}=145\text{ cm}^2[/tex]

Therefore, the total area of the margin is 145 squared cm.

Answer:

a) P ( E | F ) = 0.54545

b) P ( E | F' ) = 0

Step-by-step explanation:

Given:

- 4 Coins are tossed

- Event E exactly 2 coins shows tail

- Event F at-least two coins show tail

Find:

- Find P ( E |  F )

- Find P ( E | F prime )

Solution:

- The probability of head H and tail T = 0.5, and all events are independent

So,

                    P ( Exactly 2 T ) = ( TTHH ) + ( THHT ) + ( THTH ) + ( HTTH ) + ( HHTT) + ( HTHT)  = 6*(1/2)^4 = 0.375

                    P ( At-least 2 T ) = P ( Exactly 2 T ) + P ( Exactly 3 T ) + P ( Exactly 4 T) = 0.375 + ( HTTT) + (THTT) + (TTHT) + (TTTH) + ( TTTT)

      = 0.375 + 5*(1/2)^4 = 0.375 + 0.3125 = 0.6875

- The probabilities for each events are:

                    P ( E ) = 0.375

                    P ( F ) = 0.6875

- The Probability to get exactly two tails given that at-least 2 tails were achieved:

                    P ( E | F ) = P ( E & F ) / P ( F )

                    P ( E | F ) = 0.375 / 0.6875

                    P ( E | F ) = 0.54545

- The Probability to get exactly two tails given that less than 2 tails were achieved:

                    P ( E | F' ) = P ( E & F' ) / P ( F )

                    P ( E | F' ) = 0 / 0.6875

                    P ( E | F' ) = 0