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Love Stinks? J. Fetto, in the article "Love Stinks" (American Demographics, Vol. 25. No. 1. pp. 10-11), reports that Americans split with their significant other for many reasons-including indiscretion, infidelity, and simply "growing apart." According to the article, 35% of American adults have experienced a breakup at least once during the last 10 years. Of nine randomly selected American adults, find the probability that the number, X, who have experienced a breakup at least once during the last 10 years is

(a) exactly five; at most five; at least five.

(b) at least one; at most one.

(c) between six and eight, inclusive.

(d) Determine the probability distribution of the random variable X.

(e) Strictly speaking, why is the probability distribution that you obtained in part (d) only approximately correct? What is the exact distribution called?

Video Games. A pathological video game user (PVGU) is a video game user that averages 31 or more hours a week of gameplay.

According to the article "Pathological Video Game Use among Youths: A Two-Year Longitudinal Study" (Pediatrics, Vol. 127. No. 2, pp. 319-329) by D. Gentile et al., in 2011, about 9% of children in grades 3-8 were PVGUs. Suppose that, today, seven youths in grades 3-8 are randomly selected.

(a) Assuming that the percentage of PVGUS in grades 3-8 is the same today as it was in 2011, determine the probability distribution for the number, X, who are PVGUs.

(b) Determine and interpret the mean of X.

(c) If, in fact, exactly three of the seven youths selected are PVGUs, would you be inclined to conclude that the percentage of PVGUs in grades 3-8 has increased from the 2011 percentage? Explain your reasoning. Hint: First consider the probability $P(Xâ‰\yen 3)$.

(d) If, in fact, exactly two of the seven youths selected are PVGUs, would you be inclined to conclude the percentage of PVGUs in grades 3-8 has increased from the 2011 percentage? Explain your reasoning.

The Hypergeometric Distribution. In this exercise, we discuss the hypergeometric distribution in more detail. When sampling is done without replacement from a finite population, the hypergeometric distribution is the exact probability distribution for the number of members sampled that have a specified attribute. The hypergeometric probability formula is

$P\left(X=x\right)=\frac{\left(\begin{array}{c}Np\\ x\end{array}\right)\left(\begin{array}{c}N\left(1-p\right)\\ n-x\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)}$,

where Xdenotes the number of members sampled that have the specified attribute, Nis the population size, nis the sample size, and pis the population proportion.

To illustrate, suppose that a customer purchases 4 fuses from a shipment of 250, of which 94 % are not defective. Let a success correspond to a fuse that is not defective.

(a) Determine N, n, and p.

(b) Apply the hypergeometric probability formula to determine the probability distribution of the number of nondefective fuses that the customer gets.

Key Fact 5.6 shows that a hypergeometric distribution can be approximated by a binomial distribution, provided the sample size does not exceed 5% of the population size. In particular, you can use the binomial probability formula

$P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{\left(1-p\right)}^{n-x}$

with $n=4\mathrm{and}p=0.94$, to approximate the probability distribution of the number of nondefective fuses that the customer gets.

(c) Obtain the binomial distribution with parameters $n=4\mathrm{and}p=0.94$.

(d) Compare the hypergeometric distribution that you obtained in part (b) with the binomial distribution that you obtained in part (c).

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.

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.

Coin Tossing. A balanced dime is tossed three times. The possible outcomes can be represented as follows.

Here, for example. HHT means that the first two tosses come up heads and the third tails. Find the probability that

(a) exactly two of the three tosses come up heads.

(b) the last two tosses come up tails.

(c) all three tosses come up the same.

(d) the second toss comes up heads.

What is the difference between selecting a member at random from a finite population and taking a simple random sample of size 1?

The probability is 0.667 that the favorite in a horse race will finish in the money (first, second, or third place). In 500 horse races, roughly how many times will the favorite finish in the money?

Constract a venn diagram representing the event.

Part (a) $\left(A\&B\right)$.

Part (b)$\left(A\&B \&C\right)$.