Chapter 5: Probability and Random Variables

The Geometric Distribution. In this exercise, we discuss the geometric distribution, the probability distribution for the number of trials until the first success in Bernoulli trials. The geometric probability formula is

$P(X=x)=p{(1-p)}^{x-1},$

where Xdenotes the number of trials until the first success and pthe success probability. Using the geometric probability formula and Definition 5.9 on page 227. we can show that the mean of the random variable Xis 1/p.

To illustrate, consider the Mega Millions lottery, a multi-state jackpot draw game with a jackpot starting at $15 million and growing until someone wins. In order to play, the player selects five white numbers from the numbers 1-75 and one Mega Ball number from the numbers 1-15. Suppose that you buy one Mega Millions ticket per week. Let Xdenote the number of weeks until you win a prize.

(a) Find and interpret the probability formula for the random variable X. (Note: The probability of winning a prize with a single ticket is 0.0680.)

(b) Compute the probability that the number of weeks until you win a prize is exactly 3; at most 3: at least 3.

(c) On average, how long will it be until you win a prize?

In Exercises 5.16-5.26, express your probability answers as a decimal rounded to three places.

Housing Units. The U.S. Census Bureau publishes data on housing units in American Housing Survey for the United States. The following table provides a frequency distribution for the number of rooms in U.S. housing units. The frequencies are in thousands.

A housing unit is selected at random. Find the probability that the housing unit obtained has

(a) four rooms.

(b) more than four rooms.

(c) one or two rooms.

(d) fewer than one room.

(e) one or more rooms.

The Poisson Distribution. Another important discrete probility distribution is the Poisson distribution, named in honor of the French mathematician and physicist Simeon Poisson (1781-1840). This probability distribution is often used to model the frequency with which a specified event occurs during a particular period of time. The Poisson probability formula is

$P(X=x)=e^{-\lambda }\frac{{\lambda }^x}{x!}$.

where X is the number of times the event occurs and $\lambda$ is a parameter equal to the mean of X. The number e is the base of natural logarithms and is approximately equal to 2.7183.

To illustrate, consider the following problem: Desert Samaritan hospital, located in Mesa, Arizona, keeps records of emergency room traffic. Those records reveal that the number of patients who arrive between 6:00 P.M. and 7:00 P.M. has a Poisson distribution with parameter $\lambda =6.9$. Determine the probability that, on a given day, the number of patients who arrive at the emergency room betwee 6:00 P.M. and 7:00 P.M. will be

(a) exactly 4.

(b) at most 2.

(c)between 4 and 10, inclusive.

Concerning the equal-likelihood model of probability,

(a). what is it?

(b). how is the probability of an event found?

In Exercises 5.16-5.26, express your probability answers as a decimal rounded to three places.

Murder Victims. As reported by the Federal Bureau of Investigation in Crime in the United States, the age distribution of murder victims between 20 and 59 years old is as shown in the following table.

A murder case in which the person murdered was 59 years old is selected at random. Find the probability that the murder victim was

(a) between 40 and 44 years old, inclusive.

(b) at least 25 years old, that is, 25 years old or older.

(c) between 45 and 59 years old, inclusive.

(d) under 30 or over 54.

In Exercises 5.16-5.26, express your probability answers as a decimal rounded to three places.

Occupations in Seoul. The population of Seoul was studied in an article by B. Lee and J. McDonald, "Determinants of Commuting Time and Distance for Seoul Residents: The Impact of Family Status on the Commuting of Women" (Urban Studies, Vol. 40, No. 7, pp. 1283-1302). The authors examined the different occupations for males and females in Seoul. The table at the top of the next page is a frequency distribution of occupation type for males taking part in a survey. (Note: M = manufacturing, N = nonmanufacturing.)

If one of these males is selected at random, find the probability that his occupation is

(a) service.

(b) administrative.

(c) manufacturing.

(d) not manufacturing.

In Exercises 5.16-5.26, express your probability answers as a decimal rounded to three places.

Nobel Laureates. From Wikipedia and the article "Which Country Has the Best Brains?" from BBC News Magazine, we obtained a frequency distribution of the number of Nobel Prize winners. by country.

Suppose that a recipient of a Nobel Prize is selected at random. Find the probability that the Nobel Laureate is from

(a) Sweden.

(b) either France or Germany.

(c) any country other than the United States.

In Exercises 5.16-5.26, express your probability answers as a decimal rounded to three places.

Graduate Science Students. According to Survey of Graduate Science Engineering Students and Postdoctorates, published by the U.S. National Science Foundation, the distribution of graduate science students in doctorate-granting institutions is as follows.

Frequencies are in thousands. Note: Earth sciences include atmospheric and ocean sciences as well.

A graduate science student who is attending a doctorate-granting institution is selected at random. Determine the probability that the field of the student obtained is

(a) psychology.

(b) physical or social science.

(c) not computer science.

Family Size.A fantly is defined to be a group of two or more persons related by birth, marriage, or adoption and residing together in a household. According to Current Popularion Survey, published by the U.S. Census Bureau, the size distribution of U.S. families is as follows. Frequencies are in thousands.

A U.S. family is selected at random. Find the probability that the family obtained has

a. two persons.

b. more than three persons.

c. between one and three persons, inclusive.

d. one person.

e. one or more persons.

In Exercises 5.16-5.26, express your probability answers as a decimal rounded to three places.

Dice. Two balanced dice are rolled. Refer to Fig. 5.1 on page 198 and determine the probability that the sum of the dice is

(a) 6. (b) even.

(c). 7 or 11. (d) 2. 3, or 12.