91Ó°ÊÓ

A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) successes in the \(n\) independent trials of the experiment. $$ n=15, p=0.85, x=12 $$

Waiting in Line A Wendy's manager performed a study to determine a probability distribution for the number of people, \(X\), waiting in line during lunch. The results were as follows: $$ \begin{array}{cc|cc} x & \boldsymbol{P}(\boldsymbol{x}) & \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{x}) \\ \hline 0 & 0.011 & 7 & 0.098 \\ \hline 1 & 0.035 & 8 & 0.063 \\ \hline 2 & 0.089 & 9 & 0.035 \\ \hline 3 & 0.150 & 10 & 0.019 \\ \hline 4 & 0.186 & 11 & 0.004 \\ \hline 5 & 0.172 & 12 & 0.006 \\ \hline 6 & 0.132 & & \\ \hline \end{array} $$ (a) Verify that this is a discrete probability distribution. (b) Draw a graph of the probability distribution. Describe the shape of the distribution. (c) Compute and interpret the mean of the random variable \(X\). (d) Compute the standard deviation of the random variable \(X\). (e) What is the probability that eight people are waiting in line for lunch? (f) What is the probability that 10 or more people are waiting in line for lunch? Would this be unusual?

(a) construct a discrete probability distribution for the random variable \(X\) [Hint: \(\left.P\left(x_{i}\right)=\frac{f_{i}}{N}\right]\), (b) draw a graph of the probability distribution, (c) compute and interpret the mean of the random variable \(X,\) and \((d)\) compute the standard deviation of the random variable \(X\). $$ \begin{array}{cc} x \text { (games played) } & \text { Frequency } \\ \hline 4 & 18 \\ \hline 5 & 18 \\ \hline 6 & 20 \\ \hline 7 & 35 \end{array} $$

(a) construct a discrete probability distribution for the random variable \(X\) [Hint: \(\left.P\left(x_{i}\right)=\frac{f_{i}}{N}\right]\), (b) draw a graph of the probability distribution, (c) compute and interpret the mean of the random variable \(X,\) and \((d)\) compute the standard deviation of the random variable \(X\). Ideal Number of Children What is the ideal number of children to have in a family? The following data represent the ideal number of children for a random sample of 900 adult Americans. $$ \begin{array}{cc} x \text { (number of children) } & \text { Frequency } \\ \hline 0 & 10 \\ \hline 1 & 30 \\ \hline 2 & 520 \\ \hline 3 & 250 \\ \hline 4 & 70 \\ \hline 5 & 17 \\ \hline 6 & 3 \\ \hline \end{array} $$

A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) successes in the \(n\) independent trials of the experiment. $$ n=9, p=0.2, x \leq 3 $$

The number of hits to a website follows a Poisson process and occurs at the rate of 10 hits per minute between 7:00 P.M. and 9: 00 P.M. How long should you expect to wait before the probability of at least 1 hit to the site is \(95 \%\) ?

A life insurance company sells a \(\$ 250,000\) 1-year term life insurance policy to a 20-year-old female for \$200. According to the National Vital Statistics Report, 58(21) , the probability that the female survives the year is 0.999544 . Compute and interpret the expected value of this policy to the insurance company.

A life insurance company sells a \(\$ 250,000\) 1-year term life insurance policy to a 20-year-old male for \(\$ 350 .\) According to the National Vital Statistics Report, \(58(21),\) the probability that the male survives the year is \(0.998734 .\) Compute and interpret the expected value of this policy to the insurance company.

BlackJack is a popular casino game in which a player is dealt two cards where the value of the card corresponds to the number on the card, face cards are worth ten, and aces are worth either one or eleven. The object is to get as close to 21 as possible without going over and have cards whose value exceeds that of the dealer. A blackjack is an ace and a ten in two cards. It pays 1.5 times the bet. The dealer plays last and must draw a card with sixteen and hold with seventeen or more. The following distribution shows the winnings and probability for a \(\$ 20\) bet. In cases where the dealer and player have the same value, there is a tie (called a "push"). Source: "Examining a Gambler's Claims: Probabilistic Fact-Checking and Don Johnson's Extraordinary Winning Streak" by W.J. Hurley, Jack Brimberg, and Richard Kohar. Chance Vol. 27.1,2014 $$ \begin{array}{cc} \text { Winnings } & \text { Probability } \\ \hline 0 & 0.0982 \\ \hline \$ 30 & 0.0483 \\ \hline \$ 20 & 0.389275 \\ \hline-\$ 20 & 0.464225 \end{array} $$ (a) Compute and interpret the expected value of the game from the player's point of view. (b) Suppose over the course of one hour, a player can expect to be dealt about 40 hands. How much should a player expect to win or lose over the course of three hours?

An investment counselor calls with a hot stock tip. He believes that if the economy remains strong, the investment will result in a profit of \(\$ 50,000\). If the economy grows at a moderate pace, the investment will result in a profit of \(\$ 10,000 .\) However, if the economy goes into recession, the investment will result in a loss of \(\$ 50,000\). You contact an economist who believes there is a \(20 \%\) probability the economy will remain strong, a \(70 \%\) probability the economy will grow at a moderate pace, and a \(10 \%\) probability the economy will slip into recession. What is the expected profit from this investment?