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Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next, so that she can plan what classes to offer. Over the years, she has established the following probability distribution.

- Let $\mathrm{x}=$the number of years a student will study ballet with the teacher.

- Let $\mathrm{P}\left(\mathrm{x}\right)=$the probability that a student will study ballet $\mathrm{x}$years.

On average, how many years would you expect a child to study ballet with this teacher?

What does the column "x*P(x)" sum to and why?

You are playing a game by drawing a card from a standard deck and replacing it.If the card is face card, you win $\backslash (30$.If it is not face card,you pay $\backslash )2$.There are $12$face card in a deck of $52$cards.What is the expected value of playing game?

Use the following information to answer the next eight exercises: The Higher Education Research Institute at UCLA collected data from $203,967$incoming first-time, full-time freshmen from $270$four-year colleges and universities in the U.S. $71.3\%$of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status.

In words, define the random variable$\mathrm{X}$.

What values does the random variable X take on?

On average, how long would you expect a new hire to stay with the company?

Construct the probability distribution function (PDF).

Use the following information to answer the next eight exercises: The Higher Education Research Institute at UCLA collected data from$203,967$ incoming first-time, full-time freshmen from $270$ four-year colleges and universities in theU.S. $71.3\%$ of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status.

On average$\left(\mathrm{μ}\right),$ how many would you expect to answer yes?

A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift. For a random sample of 50 patients, the following information was obtained. Let X = the number of times a patient rings the nurse during a 12-hour shift. For this exercise, x = 0, 1, 2, 3, 4, 5. P(x) = the probability that X takes on value x. Why is this a discrete probability distribution function (two reasons)?

A trainer is teaching a dolphin to do tricks. The probability that the dolphin successfully performs the trick is 35%, and the probability that the dolphin does not successfully perform the trick is 65%. Out of 20 attempts, you want to find the probability that the dolphin succeeds 12 times. State the probability question mathematically.