# Econ 2300 ch5: Probability Distribution

## Probability Distribution

**Assignment: Ch5 Quiz**

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Exercise 5.12 METHODS AND APPLICATIONS

Suppose that the probability distribution of a random variable x can be described by the formula

P(x) = x

________________________________________

15

for each of the values x = 1, 2, 3, 4, and 5. For example, then, P(x = 2) = p(2) =2/15.

(a) Write out the probability distribution of x. (Write all fractions in reduced form.)

x 1 2 3 4 5

P(x) ________________________________________ ________________________________________ ________________________________________ ________________________________________ ________________________________________

________________________________________

(b) Show that the probability distribution of x satisfies the properties of a discrete probability distribution.(Round other answers to the nearest whole number. Leave no cells blank – be certain to enter “0” wherever required.)

P(x) ≥ for each value of x.

(c) Calculate the mean of x. (Round your answer to 3 decimal places.)

µx

(d) Calculate the variance, σ2x , and the standard deviation, σx. (Round your answer σx2 in to 3 decimal places and round answer σx in to 4 decimal places.)

σx2

σx

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Exercise 5.23 METHODS AND APPLICATIONS

Suppose that x is a binomial random variable with n = 5, p = 0.3, and q = 0.7.

(b) For each value of x, calculate p(x), and graph the binomial distribution. (Round final answers to 5 decimal places.)

p(0) = , p(1) = , p(2) = , p (3) = ,

p(4) = , p(5) =

(c) Find P(x = 3). (Round final answer to 5 decimal places.)

P(x=3)

(d) Find P(x ≤ 3). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x ≤ 3)

(e) Find P(x < 3). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x < 3) = P(x ≤ 2)

(f) Find P(x ≥ 4). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x ≥ 4)

(g) Find P(x > 2). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x > 2)

(h) Use the probabilities you computed in part b to calculate the mean, μx, the variance, σ 2x, and the standard deviation, σx, of this binomial distribution. Show that the formulas for μx , σ 2x, and σx given in this section give the same results. (Do not round intermediate calculations. Round final answers to µx and σ 2x in to 2 decimal places, and σx in to 6 decimal places.)

µx

σ2x

σx

(i) Calculate the interval [μx ± 2σx]. Use the probabilities of part b to find the probability that x will be in this interval. Hint: When calculating probability, round up the lower interval to next whole number and round down the upper interval to previous whole number. (Round your answers to 5 decimal places. A negative sign should be used instead of parentheses.)

The interval is [ , ].

P( ≤ x ≤ ) =

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MC Qu. 14 The mean of the binomial distribution is equ…

The mean of the binomial distribution is equal to:

p

np

(n) (p) (1-p)

px (1-p)n-x

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MC Qu. 25 A fair die is rolled 10 times. What is the p…

A fair die is rolled 10 times. What is the probability that an odd number (1, 3, or 5) will occur less than 3 times?

.1550

.8450

.0547

.7752

.1172

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MC Qu. 31 If n = 20 and p = .4, then the mean of the b…

If n = 20 and p = .4, then the mean of the binomial distribution is

.4

4.8

8

12

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MC Qu. 36 The probability that a given computer chip w…

The probability that a given computer chip will fail is 0.02. Find the probability that of 5 delivered chips, exactly 2 will fail.

.9039

.0000

.0922

.0038

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MC Qu. 38 In the most recent election, 19% of all elig…

In the most recent election, 19% of all eligible college students voted. If a random sample of 20 students were surveyed:

Find the probability that exactly half voted in the election.

.4997

.0014

.0148

.0000

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MC Qu. 39 In the most recent election, 19% of all elig…

In the most recent election, 19% of all eligible college students voted. If a random sample of 20 students were surveyed:

Find the probability that none of the students voted.

.0148

.4997

.0014

.0000

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MC Qu. 55 For a random variable X, the mean value of t…

For a random variable X, the mean value of the squared deviations of its values from their expected value is called its

Standard Deviation

Mean

Probability

Variance

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MC Qu. 62 If the probability distribution of X is:&nbs…

If the probability distribution of X is:

What is the expected value of X?

2.25

2.24

1.0

5.0

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MC Qu. 63 If the probability distribution of X is:&nbs…

If the probability distribution of X is:

What is the variance of X?

5.0

→

1.0

2.25

2.24

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MC Qu. 66 A vaccine is 95 percent effective. What is t…

A vaccine is 95 percent effective. What is the probability that it is not effective for, more than one out of 20 individuals?

.3774

.7359

→

.2641

.3585

P(X ≥ 2) = 1 – [P(X = 0) + p(X = 1)]

P(X ≥ 2) = 1 – [(.3585) + (.3774)] = .2641

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MC Qu. 67 If the probability of a success on a single …

If the probability of a success on a single trial is .2, what is the probability of obtaining 3 successes in 10 trials if the number of successes is binomial?

.1074

.2013

.5033

.0031

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MC Qu. 78 For a binomial process, the probability of s…

For a binomial process, the probability of success is 40% and the number of trials is 5.

Find the expected value.

5.0

1.1

1.2

2.0

E[X] = (5) (.40) = 2

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MC Qu. 79 For a binomial process, the probability of s…

For a binomial process, the probability of success is 40% and the number of trials is 5.

Find the variance.

1.1

→

1.2

5.0

2.0

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MC Qu. 80 For a binomial process, the probability of s…

For a binomial process, the probability of success is 40% and the number of trials is 5.

Find the standard deviation.

1.1

5.0

2.0

1.2

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MC Qu. 82 For a binomial process, the probability of s…

For a binomial process, the probability of success is 40% and the number of trials is 5.

Find P(X > 4).

.0102

.0778

.0870

.3370

P(X = 5) = (.4)5 = .0102

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MC Qu. 92 If X has the probability distribution%…

If X has the probability distribution

compute the expected value of X.

0.5

0.7

1.0

0.3

E[X] = -1(.2) + 0(.3) + 1(.5) = .3

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MC Qu. 93 If X has the probability distribution%…

If X has the probability distribution

compute the expected value of X.

1.3

2.4

1.0

1.8

E[X] = (-2) (.2) + (-1) (.2) + (1) (.2) + (2) (.2) + (9) (.2) = 1.8

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MC Qu. 95 X has the following probability distribution…

X has the following probability distribution P(X):

Compute the variance value of X.

1.58

.625

→

.850

.955

E[X] = (1) (.1) + (2) (.5) + (3) (.2) + (4) (.2) = 2.5

= (1 – 2.5)2 (.1) + (2 – 2.5)2 (.5) + (3 – 2.5)2 (.2) + (4 – 2.5)2 (.2) = .850

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MC Qu. 99 Consider the experiment of tossing a fair co…

Consider the experiment of tossing a fair coin three times and observing the number of heads that result (X = number of heads).

Determine the expected number of heads.

1.1

1.5

1.0

2.0

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MC Qu. 100 Consider the experiment of tossing a fair co…

Consider the experiment of tossing a fair coin three times and observing the number of heads that result (X = number of heads).

What is the variance for this distribution?

→

0.75

0.87

1.22

1.5

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MC Qu. 102 Consider the experiment of tossing a fair co…

Consider the experiment of tossing a fair coin three times and observing the number of heads that result (X = number of heads).

If you were asked to play a game in which you tossed a fair coin three times and were given $2 for every head you threw, how much would you expect to win on average?

$6

→

$3

$9

$2

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MC Qu. 104 According to data from the state blood progr…

According to data from the state blood program, 40% of all individuals have group A blood. If six (6) individuals give blood, find the probability that None of the individuals has group A blood?

.0467

.4000

.0041

.0410

View Hint #1

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MC Qu. 105 According to data from the state blood progr…

According to data from the state blood program, 40% of all individuals have group A blood. If six (6) individuals give blood, find the probability that Exactly three of the individuals has group A blood?

.4000

.2765

.5875

.0041

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MC Qu. 106 According to data from the state blood progr…

According to data from the state blood program, 40% of all individuals have group A blood. If six (6) individuals give blood, find the probability that At least 3 of the individuals have group A blood.

.4557

.8208

.1792

.5443

P(x ≥ 3) = P(x = 3) + p(x = 4) + p(x = 5) + p(x = 6) = .4557

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MC Qu. 113 An important part of the customer service re…

An important part of the customer service responsibilities of a cable company relates to the speed with which trouble in service can be repaired. Historically, the data show that the likelihood is 0.75 that troubles in a residential service can be repaired on the same day. For the first five troubles reported on a given day, what is the probability that fewer than two troubles will be repaired on the same day?

.0010

.0146

→

.0156

.6328

P(x < 2) = P(x = 0) + P(x = 1) = .0156

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MC Qu. 116 The Post Office has established a record in …

The Post Office has established a record in a major Midwestern city for delivering 90% of its local mail the next working day. If you mail eight local letters, what is the probability that all of them will be delivered the next day.

1.0

.5695

→

.4305

.8131

P(x = 8) = .4305

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MC Qu. 117 The Post Office has established a record in …

The Post Office has established a record in a major Midwestern city for delivering 90% of its local mail the next working day. Of the eight, what is the average number you expect to be delivered the next day?

4.0

2.7

3.6

7.2

= np = (8) (.9) = 7.2

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MC Qu. 118 The Post Office has established a record in …

The Post Office has established a record in a major Midwestern city for delivering 90% of its local mail the next working day. Calculate the standard deviation of the number delivered when 8 local letters are mailed.

.72

→

.85

2.83

2.68

Σ = = = = .85

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MC Qu. 124 A large disaster cleaning company estimates …

A large disaster cleaning company estimates that 30% of the jobs it bids on are finished within the bid time. Looking at a random sample of 8 jobs that is has contracted calculate the mean number of jobs completed within the bid time.

2.0

5.6

4.0

2.4

= np = 8(.3) = 2.4

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MC Qu. 125 A large disaster cleaning company estimates …

A large disaster cleaning company estimates that 30% of the jobs it bids on are finished within the bid time. Looking at a random sample of 8 jobs that is has contracted find the probability that x (number of jobs finished on time) is within one standard deviation of the mean.

→

.5506

.6867

.8844

.7483

σ = √npq = 1.3. P(µ+/-σ) = P(2.4+/-1.3) = P(1.1≤X≤3.7) = P(2≤X≤3) = 0.2965+0.2541 = 0.5506