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X ~ _____(_____,_____)

What values does X take on?

Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given in Table 4.31.

a. In words, define the random variable X.

b. What does it mean that the values zero, one, and two are not included for x in the PDF?

What is the probability the baker will sell more than one batch? $P(x>1)=\_\_\_\_\_\_\_$

A theater group holds a fund-raiser. It sells 100 raffle tickets for \(5 apiece. Suppose you purchase four tickets. The prize is two passes to a Broadway show, worth a total of \)150.

a. What are you interested in here?

b. In words, define the random variable X.

c. List the values that X may take on.

d. Construct a PDF.

e. If this fund-raiser is repeated often and you always purchase four tickets, what would be your expected average winnings per raffle?

A game involves selecting a card from a regular $52$-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails.

鈥� If the card is a face card, and the coin lands on heads, you win $\backslash (6$

鈥� If the card is a face card, and the coin lands on tails, you win $\backslash )2$

鈥� If the card is not a face card, you lose $\$2,$no matter what the coin shows.

a. Find the expected value for this game (expected net gain or loss).

b. Explain what your calculations indicate about your long-term average profits and losses on this game.

c. Should you play this game to win money?

You buy a lottery ticket to a lottery that costs $\backslash (10$ per ticket. There are only $100$ tickets available to be sold in this lottery. In this lottery there are one $\backslash )500$ prize, two $\backslash (100$ prizes, and four $\backslash )25$ prizes. Find your expected gain or loss.

Suppose that you are offered the following 鈥渄eal.鈥� You roll a die. If you roll a six, you win $\backslash (10.$ If you roll a four or five, you win $\backslash )5.$ If you roll a one, two, or three, you pay $\$6.$

a. What are you ultimately interested in here (the value of the roll or the money you win)?

b. In words, define the Random VariableX.

c. List the values that X may take on.

d. Construct a PDF.

e. Over the long run of playing this game, what are your expected average winnings per game?

f. Based on numerical values, should you take the deal? Explain your decision in complete sentences.

A venture capitalist, willing to invest $\backslash (1,000,000$, has three investments to choose from. The first investment, a software company, has a $10\%$chance of returning $\backslash )5,000,000$profit, a $30\%$chance of returning $\backslash (1,000,000$profit, and a $60\%$chance of losing the million dollars. The second company, a hardware company, has a $20\%$chance of returning $\backslash )3,000,000$profit, a $40\%$chance of returning $\backslash (1,000,000$profit, and a $40\%$chance of losing the million dollars. The third company, a biotech firm, has a $10\%$chance of returning $\backslash )6,000,000$profit, a $70\%$of no profit or loss, and a $20\%$chance of losing the million dollars.

a. Construct a PDF for each investment.

b. Find the expected value for each investment.

c. Which is the safest investment? Why do you think so?

d. Which is the riskiest investment? Why do you think so?

e. Which investment has the highest expected return, on average?

Suppose that $20,000$ married adults in the United States were randomly surveyed as to the number of children they have. The results are compiled and are used as theoretical probabilities. Let X = the number of children married people have.

a. Find the probability that a married adult has three children.

b. In words, what does the expected value in this example represent?

c. Find the expected value.

d. Is it more likely that a married adult will have two to three children or four to six children? How do you know?