Deck 5: Probability Distributions

Question 1

Identify each of the following as a discrete or continuous random variable.
Distance traveled for a migrating bird.

Answer 1

Continuous

Question 2

Each of the judges in a popular dancing contest assign point from 3 through 5 to each participating couple. Let X be the sum of the two scores assigned to a particular couple. List the possible values for the sum of the two scores and also the pairs of scores that result in that sum.

Answer 2

Question 3

A rent a car company has two types of vehicles: compact (C) and full size (F). For two consecutive months, the manager recorded the type with the highest customer acceptance. List the possible outcomes, in alphabetical order; each outcome must be followed by the number of months that a full size car has the highest acceptance. That is: Letter-Letter Number

Answer 3

C-C 0, C-F 1, C-S 0, F-C 1, F-F 2, F-S 1, S-C 0, S-F 1, S-S 0

Question 4

A junior college student is planning her spring semester; there are six courses available and she must choose four. The courses are:

A) Make a list, in alphabetical order, of the outcomes. (Use B, D, E, J, M, and P to denote the five courses).
B) Let Y be the number of courses in Group 2 included.
C) Use the information obtained in (A) and (B) to complete the following table.

Question 5

The events corresponding to the distinct values of a random variable X are incompatible.

Question 6

A random variable can be ______ or continuous.

Question 7

If X represents the number of times that a head is followed by a tail obtained in four tosses of a fair coin, find the probability distribution of X. Write the probabilities using decimal numbers.

Question 8

Let X be the number of vowels in 50 words taken at random from a dictionary. If f(2) = .12 , f(3) = .2, f(4) = .15, and f(5) = .17, find f(6).

A) .53
B) .48
C) .36
D) .56

Question 9

Probability distributions are sample based and then susceptible to variation on different occasions of sampling.

Question 10

Let the random variable X represent the sum of points in a tile taken from the dominoes set shown below.
What is the probability of 9? Write your answer as a fraction reduced to lowest terms.

Question 11

Determine how many of the following are a legitimate probability distribution.

A) 0
B) 3
C) 1
D) 2

Question 12

Consider the probability distribution given by the function
for Find f(5). Round your answer to three decimal places.

Question 13

The probability distribution of X is given by the function
for x = 2, 3, 4, 5.
Find P[X = 4 ]. Round your answer to two decimal places.

Question 14

Your scores (0-50) in the six homework assignments for a course are: 46 $\quad$ 46 $\quad$ 46 $\quad$ 50 $\quad$ 50 $\quad$ 50. The instructor will add the scores of two randomly selected assignment, this number will be your homework grade. Let X denote the sum of the scores of the two selected assignment. Complete the following table of probability distribution of X. Round the probabilities to two decimal places when necessary.

Question 15

A probability distribution is partially given in the following table with the additional information that the even values of X are equally likely. Determine the missing entries in the table.

Question 16

Let X be the number of vowels used in the name of each U.S. state. Use the following table to calculate:

A)

B)

Question 17

With X denoting the number of 1's or 2's in two rolls of a fair die, calculate the mean of X. Round your answer to two decimal places.

Question 18

Use the information in the table below to calculate the expected value of x. Round your answer to three decimal places.

Question 19

Calculate the variance of the distribution of X

A) 2.08
B) 2.16
C) 2.3
D) 2.36

Question 20

Calculate the standard deviation of the distribution of X.

A) 1.84
B) 1.62
C) 1.52
D) 1.48

Question 21

The variance rather than the standard deviation, is the appropriate measure of spread.

Question 22

Given the following probability distribution, find:

A) E(X)
B) $\sigma$ ²
C) $\sigma$

Question 23

Is the model of Bernoulli trials plausible in the following situation?
Students in a literature course must read a book, the time spent reading the book is recorded.

Question 24

In Bernoulli Trials, if P(S) = p, then P(F) = 1 - p.

Question 25

If elements are sampled from a small dichotomous population at random and ______ replacement, the conditions for Bernoulli trials are satisfied.

Question 26

A cookie jar contains 42 cookies of which 10 are oatmeal, 18 are sugar, and 14 are macaroon. Consider 5 successive draws of one cookie from the jar and suppose the selected cookie is returned to the jar. If the appearance of an oatmeal cookie is the event of interest, determine the numerical value of p. Round your answer to two decimal places

Question 27

Consider Bernoulli trials with success probability p = 0.7.

A) Find the probability that three trials result in all failures.
B) Find the probability that the first failure occurs in the third trial.
Round your answer to three decimal places.

Question 28

The accompanying table shows the percentages of students in the state university system when classified according to gender and status as residents of that state.
Suppose that a selection of a person is considered a trial and it is a success when the person selected is a resident of that state.
Three persons are selected at random. What is the probability that none is a state resident? Round your answer to three decimal places.

Question 29

The accompanying table shows the percentages of students in the state university system when classified according to gender and status as residents of that state.
Suppose that a selection of a person is considered a trial and it is a success when the person selected is a resident of that state.
Five females are selected at random. What is the probability that none is a state resident? Round your answer to three decimal places.

Question 30

The accompanying table shows the percentages of students in the state university system when classified according to gender and status as residents of that state.
Suppose that a selection of a person is considered a trial and it is a success when the person selected is a resident of that state.
Two males and two females are selected at random. What is the probability that none is a state resident? Round your answer to three decimal places.

Question 31

Find the probability of 5 successes in 7 Bernoulli trials with success probability 0.7 for each trial. Round your answer to three decimal places.

Question 32

If p is the probability of success and q = 1 - p, then the probability distribution of X for four Bernoulli trials is:

Question 33

If p is the probability of success and q is the probability of failure, then the probability of X = 5 for 8 Bernoulli trials is:

A) $56 p^{5}$

B) $56 p^{5} q^{3}$

C) $p^{5} q^{3}$

D) $126 p^{5} q^{3}$

Question 34

Use the Bernoulli Table (Appendix B, Table 2) to find P[X $\ge$ 6] for n = 8 and p = 0.7.

A) .255
B) .745
C) .265
D) .735

Question 35

Consider the binomial distribution with 17 trials and success probability 0.68. Find its:

A) Mean
B) Variance
C) Standard Deviation

Question 36

A shooting athlete using a shotgun has an 81% chance of success (shot on target). Let X denote the number of successes in the next six rounds. Assuming the results for different rounds are independent, calculate P[X = 3 or 5]. Round your answer to two decimal places.

Question 37

About 78% of children in a neighborhood have a bicycle. Suppose n = 4 children are randomly selected. Find the probability that at most two have a bicycle. Round your answer to two decimal places.

Question 38

Suppose 30% of the urban population is 18 years old or younger. Five residents of that city are selected at random; find the probability that two of the selected five persons are 18 years old or younger.

A) .13
B) .18
C) .26
D) .31

Question 39

Given that n = 25 and p₀ = .35, find:

A) The lower control limit.
B) The upper control limit.

Question 40

Given that the lower control limit is 0.12 and p₀ = .5, find the value of n.

A) 5
B) 11
C) 16
D) Cannot be determined

Question 41

Use the Poisson Table (Appendix A, Table 3) to find P [ X $\ge$ 4] for rate parameter m = 2.4

A) .779
B) 0.430
C) .221
D) .096

Question 42

Consider a Poisson random variable with rate parameter m = 4. Find its:
(a) Mean
(b) Variance
(c) Standard Deviation

Question 43

Is the Poisson distribution a suitable approximation for to the binomial distribution with n = 55 and p = .03

Question 44

The probability of red coming up on a roulette wheel that has 18 red, 18 black, and 2 green slots is 18/38 or about .474. If you bet on red a large number of times, what proportion of times will you win. Explain your reasoning.

Question 45

John and two of his friends have one-on-one basketball tournaments almost every week. Each person plays each other person once. Let X be the number of games that John wins next week X. Based on the results of the games over the past year, John constructs the following probability distribution for X.

Find
(a) The expected number of wins.
(b) The standard deviation of the number of wins.
(c) Explain how John should use the results of previous games to construct the tabled probability distribution.

Question 46

Given the following probability distribution for a random variable X,

Find
(a) E ( X )
(b) $\sigma$ ² = Var ( X )
(c) standard deviation of X.

Question 47

You are traveling to Stockholm, Sweden next month when the average temperature is -1.5 degrees Celsius. Treating this as the expected value of a random variable X, use the properties of expectation to determine the expected value of this temperature, Y , in degrees Fahrenheit. Recall that

Question 48

About 75% of dog owners buy holiday presents for their dogs. Sup- pose n = 3 dog owners are randomly selected. Use the formula for the binomial distribution to find the probability of:
(a) Two or more buy their dog holiday presents.
(b) At most two buy their dog holiday presents.
(c) Find the expected number of persons, in the sample, who buy their dog holiday presents.

Question 49

The manager of a satellite TV service states that the rate of
interruption of services is .4 per month. Use the Poisson distribution in Table 3 of Appendix A to find
(a) Zero interruptions in one month.
(b) One or more interruptions in one month.
(c) Exact two interruptions in 3 consecutive months.