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An instant lottery ticket costs \(\$ 2\). Out of a total of 10,000 tickets printed for this lottery, 1000 tickets contain a prize of \(\$ 5\) each, 100 tickets have a prize of \(\$ 10\) each, 5 tickets have a prize of \(\$ 1000\) each, and 1 ticket has a prize of \(\$ 5000 .\) Let \(x\) be the random variable that denotes the net amount a player wins by playing this lottery. Write the probability distribution of \(x\). Determine the mean and standard deviation of \(x\). How will you interpret the values of the mean and standard deviation of \(x ?\)

A ski patrol unit has nine members available for duty, and two of them are to be sent to rescue an injured skier. In how many ways can two of these nine members be selected? Now suppose the order of selection is important. How many arrangements are possible in this case?

According to a Harris Interactive poll, \(52 \%\) of American college graduates have Facebook accounts (http://www.harrisinteractive.com/harris_poll/pubs/Harris_Poll \(200904_{-16}\).pdf). Suppose that this result. is true for the current population of American college graduates. a. Let \(x\) be a binomial random variable that denotes the number of American college graduates in a random sample of 15 who have Facebook accounts. What are the possible values that \(x\) can assume? b. Find the probability that exactly 9 American college graduates in a sample of 15 have Facebook accounts.

In a 2009 poll of adults 18 years and older, (BBMG Conscious Consumer Report) about half of them said that despite tough economic times, they are willing to pay more for products that have social and environmental benefits. Suppose that \(50 \%\) of all such adults currently hold this view. Suppose that a random sample of 20 such adults is selected. Use the binomial probabilities table (Table I of Appendix \(\mathrm{C}\) ) or technology to find the probability that the number of adults in this sample who hold this opinion is a. at most 7 b. at least 13 c. 12 to 15

A fast food chain store conducted a taste survey before marketing a new hamburger. The results of the survey showed that \(70 \%\) of the people who tried this hamburger liked it. Encouraged by this result, the company decided to market the new hamburger. Assume that \(70 \%\) of all people like this hamburger. On a certain day, eight customers bought it for the first time. a. Let \(x\) denote the number of customers in this sample of eight who will like this hamburger. Using the binomial probabilities table, obtain the probability distribution of \(x\) and draw a graph of the probability distribution. Determine the mean and standard deviation of \(x\). b. Using the probability distribution of part a, find the probability that exactly three of the eight customers will like this hamburger.

What are the conditions that must be satisfied to apply the Poisson probability distribution?

Let \(x\) be the number of cars that a randomly selected auto mechanic repairs on a given day. The following table lists the probability distribution of \(x\). $$ \begin{array}{l|ccccc} \hline x & 2 & 3 & 4 & 5 & 6 \\ \hline P(x) & .05 & .22 & .40 & 23 & .10 \\ \hline \end{array} $$ Find the mean and standard deviation of \(x\), Give a brief interpretation of the value of the mean.

Scott offers you the following game: You will roll two fair dice. If the sum of the two numbers obtained is \(2,3,4,9,10,11\), or 12, Scott will pay you \(\$ 20\). However, if the sum of the two numbers is 5 , 6,7, or 8 , you will pay Scott \(\$ 20\). Scott points out that you have seven winning numbers and only four losing numbers. Is this game fair to you? Should you accept this offer? Support your conclusion with appropriate calculations.

Two teams, \(\mathrm{A}\) and \(\mathrm{B}\), will play a best-of-seven series, which will end as soon as one of the teams wins four games. Thus, the series may end in four, five, six, or seven games. Assume that each team has an equal chance of winning each game and that all games are independent of one another. Find the following probabilities. a. Team A wins the series in four games. b. Team A wins the series in five games. c. Seven games are required for a team to win the series.

A high school history teacher gives a 50 -question multiple-choice examination in which each question has four choices. The scoring includes a penalty for guessing. Each correct answer is worth I point, and each wrong answer costs \(1 / 2\) point. For example, if a student answers 35 questions correctly, 8 questions incorrectly, and does not answer 7 questions, the total score for this student will be \(35-(1 / 2)(8)=31\) a. What is the expected score of a student who answers 38 questions correctly and guesses on the other 12 questions? Assume that the student randomly chooses one of the four answers for each of the 12 guessed questions. b. Does a student increase his expected score by guessing on a question if he has no idea what the correct answer is? Explain. c. Does a student increase her expected score by guessing on a question for which she can eliminate one of the wrong answers? Explain.