CALCU-Q1. P(A)=0.10, P(B A)=0.39, and P(B A)=0.3

Solution by Steps

step 1

To find

P(\bar{A})

, we use the formula

P(\bar{A}) = 1 - P(A)

. Given

P(A) = 0.10

, we have:

P(\bar{A}) = 1 - 0.10 = 0.90

step 2

To find

P(\bar{B} \mid A)

, we use the formula

P(\bar{B} \mid A) = 1 - P(B \mid A)

. Given

P(B \mid A) = 0.39

, we have:

P(\bar{B} \mid A) = 1 - 0.39 = 0.61

step 3

To find

P(\bar{B} \mid \bar{A})

, we use the formula

P(\bar{B} \mid \bar{A}) = 1 - P(B \mid \bar{A})

. Given

P(B \mid \bar{A}) = 0.39

, we have:

P(\bar{B} \mid \bar{A}) = 1 - 0.39 = 0.61

step 4

To find

P(B)

, we use the law of total probability:

P(B) = P(B \mid A)P(A) + P(B \mid \bar{A})P(\bar{A})

. Substituting the known values:

P(B) = 0.39 \times 0.10 + 0.39 \times 0.90 = 0.039 + 0.351 = 0.39

step 5

To find

P(\bar{B})

, we use the formula

P(\bar{B}) = 1 - P(B)

. Thus,

P(\bar{B}) = 1 - 0.39 = 0.61

step 6

To find

P(A \mid B)

, we use Bayes' theorem:

P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B)}

. Substituting the known values:

P(A \mid B) = \frac{0.39 \times 0.10}{0.39} = 0.10

step 7

To find

P(\bar{A} \mid B)

, we use Bayes' theorem:

P(\bar{A} \mid B) = \frac{P(B \mid \bar{A})P(\bar{A})}{P(B)}

. Substituting the known values:

P(\bar{A} \mid B) = \frac{0.39 \times 0.90}{0.39} = 0.90

step 8

To find

P(A \mid \bar{B})

, we use Bayes' theorem:

P(A \mid \bar{B}) = \frac{P(\bar{B} \mid A)P(A)}{P(\bar{B})}

. Substituting the known values:

P(A \mid \bar{B}) = \frac{0.61 \times 0.10}{0.61} = 0.10

step 9

To find

P(\bar{A} \mid \bar{B})

, we use Bayes' theorem:

P(\bar{A} \mid \bar{B}) = \frac{P(\bar{B} \mid \bar{A})P(\bar{A})}{P(\bar{B})}

. Substituting the known values:

P(\bar{A} \mid \bar{B}) = \frac{0.61 \times 0.90}{0.61} = 0.90

Answer

P(\bar{A}) = 0.90, P(\bar{B} \mid A) = 0.61, P(\bar{B} \mid \bar{A}) = 0.61, P(B) = 0.39, P(\bar{B}) = 0.61, P(A \mid B) = 0.10, P(\bar{A} \mid B) = 0.90, P(A \mid \bar{B}) = 0.10, P(\bar{A} \mid \bar{B}) = 0.90

Key Concept

Conditional probability and Bayes' theorem are essential for finding probabilities of events based on given conditions.

Explanation

The calculations show how to derive various probabilities using the provided values and the relationships between events A and B. The results indicate that A and B are independent events since

P(B \mid A) = P(B \mid \bar{A})

.

Julie Myers, a graduating senior in accounting, is preparing for an interview with a Big Four accounting firm. Before the interview, she sets her chances of getting an offer at 50%. Then, on thinking about her friends who have interviewed and gotten offers from this firm, she realizes that of the people who received offers, 95% had good interviews. On the other hand, of those who did not receive offers, 75% said they had good interviews. If Julie Myers has a good interview, what are her chances of receiving an offer?

Solution by Steps

step 1

Let A be the event that Julie receives an offer and B be the event that she has a good interview. We need to find

P(A \mid B)

. Using Bayes' theorem, we have:

P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)}

step 2

We know from the problem that

P(A) = 0.50

and

P(B \mid A) = 0.95

. Now we need to find

P(B)

. We can use the law of total probability:

P(B) = P(B \mid A) P(A) + P(B \mid \bar{A}) P(\bar{A})

step 3

We know

P(B \mid \bar{A}) = 0.75

and

P(\bar{A}) = 1 - P(A) = 0.50

. Substituting these values, we get:

P(B) = (0.95)(0.50) + (0.75)(0.50) = 0.475 + 0.375 = 0.85

step 4

Now we can substitute back into Bayes' theorem:

P(A \mid B) = \frac{(0.95)(0.50)}{0.85} = \frac{0.475}{0.85} \approx 0.5588

Answer

Julie's chances of receiving an offer given that she has a good interview are approximately 55.88%.

Key Concept

Bayes' theorem is used to find conditional probabilities based on prior knowledge.

Explanation

The calculation shows how to update the probability of receiving an offer based on the information about having a good interview.

Solution by Steps

step 1

To find the expected value

E(X)

for distribution a, we use the formula:

E(X) = 1 \times 0.05 + 2 \times 0.45 + 3 \times 0.30 + 4 \times 0.20

step 2

Calculating

E(X)

:

E(X) = 0.05 + 0.90 + 0.90 + 0.80 = 2.65

step 3

To find the variance

Var(X)

, we first calculate

E(X^2)

:

E(X^2) = 1^2 \times 0.05 + 2^2 \times 0.45 + 3^2 \times 0.30 + 4^2 \times 0.20

step 4

Calculating

E(X^2)

:

E(X^2) = 0.05 + 1.80 + 2.70 + 3.20 = 7.75

step 5

Now, we find the variance:

Var(X) = E(X^2) - (E(X))^2 = 7.75 - (2.65)^2

step 6

Calculating

Var(X)

:

Var(X) = 7.75 - 7.0225 = 0.7275

step 7

The standard deviation

SD(X)

is the square root of the variance:

SD(X) = \sqrt{Var(X)} = \sqrt{0.7275} \approx 0.853

Answer

For distribution a:

E(X) = 2.65

,

Var(X) \approx 0.7275

,

SD(X) \approx 0.853

Key Concept

Expected value, variance, and standard deviation in probability distributions

Explanation

The expected value gives the average outcome, variance measures the spread of the distribution, and standard deviation provides a measure of the average distance from the mean.

--- Now, let's move on to distribution b.

Solution by Steps

step 1

To find the expected value

E(X)

for distribution b, we use the formula:

E(X) = -20 \times 0.13 + 0 \times 0.58 + 100 \times 0.29

step 2

Calculating

E(X)

:

E(X) = -2.6 + 0 + 29 = 26.4

step 3

To find the variance

Var(X)

, we first calculate

E(X^2)

:

E(X^2) = (-20)^2 \times 0.13 + 0^2 \times 0.58 + 100^2 \times 0.29

step 4

Calculating

E(X^2)

:

E(X^2) = 400 \times 0.13 + 0 + 10000 \times 0.29 = 52 + 0 + 2900 = 2952

step 5

Now, we find the variance:

Var(X) = E(X^2) - (E(X))^2 = 2952 - (26.4)^2

step 6

Calculating

Var(X)

:

Var(X) = 2952 - 696.96 = 2255.04

step 7

The standard deviation

SD(X)

is the square root of the variance:

SD(X) = \sqrt{Var(X)} = \sqrt{2255.04} \approx 47.5

Answer

For distribution b:

E(X) = 26.4

,

Var(X) \approx 2255.04

,

SD(X) \approx 47.5

Key Concept

Expected value, variance, and standard deviation in probability distributions

Explanation

The expected value indicates the average outcome, while variance and standard deviation measure the variability of the outcomes in the distribution.

--- Now, let's analyze distribution c.

Solution by Steps

step 1

To find the expected value

E(X)

for distribution c, we use the formula:

E(X) = 0 \times 0.368 + 1 \times 0.632

step 2

Calculating

E(X)

:

E(X) = 0 + 0.632 = 0.632

step 3

To find the variance

Var(X)

, we first calculate

E(X^2)

:

E(X^2) = 0^2 \times 0.368 + 1^2 \times 0.632

step 4

Calculating

E(X^2)

:

E(X^2) = 0 + 0.632 = 0.632

step 5

Now, we find the variance:

Var(X) = E(X^2) - (E(X))^2 = 0.632 - (0.632)^2

step 6

Calculating

Var(X)

:

Var(X) = 0.632 - 0.399424 = 0.232576

step 7

The standard deviation

SD(X)

is the square root of the variance:

SD(X) = \sqrt{Var(X)} = \sqrt{0.232576} \approx 0.482

Answer

For distribution c:

E(X) = 0.632

,

Var(X) \approx 0.232576

,

SD(X) \approx 0.482

Key Concept

Expected value, variance, and standard deviation in probability distributions

Explanation

The expected value represents the average outcome, while variance and standard deviation quantify the dispersion of the outcomes in the distribution.

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