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In how many ways can a sample (without replacement) of 5 items be selected from a population of 15 items?

A random sample of 250 juniors majoring in psychology or communication at a large university is selected. These students are asked whether or not they are happy with their majors. The following table gives the results of the survey. Assume that none of these 250 students is majoring in both areas. $$ \begin{array}{lcc} \hline & \text { Happy } & \text { Unhappy } \\ \hline \text { Psychology } & 80 & 20 \\ \text { Communication } & 115 & 35 \\ \hline \end{array} $$ a. If one student is selected at random from this group, find the probability that this student is i. happy with the choice of major ii. a psychology major iii. a communication major given that the student is happy with the choice of major iv. unhappy with the choice of major given that the student is a psychology major v. a psychology major and is happy with that major vi. a communication major \(o r\) is unhappy with his or her major b. Are the events "psychology major" and "happy with major" independent? Are they mutually exclusive? Explain why or why not.

A random sample of 250 adults was taken, and they were asked whether they prefer watching sports or opera on television. The following table gives the two-way classification of these adults. $$ \begin{array}{lcc} \hline & \begin{array}{c} \text { Prefer Watching } \\ \text { Sports } \end{array} & \begin{array}{c} \text { Prefer Watching } \\ \text { Opera } \end{array} \\ \hline \text { Male } & 96 & 24 \\ \text { Female } & 45 & 85 \\ \hline \end{array} $$ a. If one adult is selected at random from this group, find the probability that this adult i. prefers watching opera ii. is a male iii. prefers watching sports given that the adult is a female iv. is a male given that he prefers watching sports \(\mathrm{v} .\) is a female and prefers watching opera vi. prefers watching sports or is a male b. Are the events "female" and "prefers watching sports" independent? Are they mutually exclusive? Explain why or why not.

A random sample of 400 college students was asked if college athletes should be paid. The following table gives a two-way classification of the responses. $$ \begin{array}{lcc} \hline & \text { Should Be Paid } & \text { Should Not Be Paid } \\ \hline \text { Student athlete } & 90 & 10 \\ \text { Student nonathlete } & 210 & 90 \\ \hline \end{array} $$ a. If one student is randomly selected from these 400 students, find the probability that this student i. is in favor of paying college athletes ii. favors paying college athletes given that the student selected is a nonathlete iii. is an athlete and favors paying student athletes iv. is a nonathlete \(o r\) is against paying student athletes b. Are the events "student athlete" and "should be paid" independent? Are they mutually exclusive? Explain why or why not.

Powerball is a game of chance that has generated intense interest because of its large jackpots. To play this game, a player selects five different numbers from 1 through 59, and then picks a Powerball number from 1 through \(39 .\) The lottery organization randomly draws 5 different white balls from 59 balls numbered 1 through 59 , and then randomly picks a Powerball number from 1 through \(39 .\) Note that it is possible for the Powerball number to be the same as one of the first five numbers. a. If a player's first five numbers match the numbers on the five white balls drawn by the lottery organization and the player's Powerball number matches the Powerball number drawn by the lottery organization, the player wins the jackpot. Find the probability that a player who buys one ticket will win the jackpot. (Note that the order in which the five white balls are drawn is unimportant.) b. If a player's first five numbers match the numbers on the five white balls drawn by the lottery organization, the player wins about \(\$ 200,000\). Find the probability that a player who buys one ticket will win this prize.

A screening test for a certain disease is prone to giving false positives or false negatives. If a patient being tested has the disease, the probability that the test indicates a (false) negative is \(.13 .\) If the patient does not have the disease, the probability that the test indicates a (false) positive is .10. Assume that \(3 \%\) of the patients being tested actually have the disease. Suppose that one patient is chosen at random and tested. Find the probability that a. this patient has the disease and tests positive b. this patient does not have the disease and tests positive c. this patient tests positive d. this patient has the disease given that he or she tests positive (Hint: A tree diagram may be helpful in part c.)

A pizza parlor has 12 different toppings available for its pizzas, and 2 of these toppings are pepperoni and anchovies. If a customer picks 2 toppings at random, find the probability that a. neither topping is anchovies b. pepperoni is one of the toppings

The Big Six Wheel (or Wheel of Fortune) is a casino and carnival game that is well known for being a big money maker for the casinos. The wheel has 54 equally likely slots (outcomes) on it. The slot that pays the largest amount of money is called the joker. If a player bets on the joker, the probability of winning is \(1 / 54\). The outcome of any given play of this game (a spin of the wheel) is independent of the outcomes of previous plays. a. Find the probability that a player who always bets on joker wins for the first time on the 15 th play of the game. b. Find the probability that it takes a player who always bets on joker more than 70 plays to win for the first time.