4.9 Homework  (Page 3/7)

The chance of having an extra fortune in a fortune cookie is about 3%. Given a bag of 144 fortune cookies, we are interested in the number of cookies with an extra fortune. Two distributions may be used to solve this problem. Use one distribution to solve the problem.

• d How many cookies do we expect to have an extra fortune?
• e Find the probability that none of the cookies have an extra fortune.
• f Find the probability that more than 3 have an extra fortune.
• g As $n$ increases, what happens involving the probabilities using the two distributions? Explain in complete sentences.

There are two games played for Chinese New Year and Vietnamese New Year. They are almost identical. In the Chinese version, fair dice with numbers 1, 2, 3, 4, 5, and 6 are used, along with a board with those numbers. In the Vietnamese version, fair dice with pictures of a gourd, fish, rooster, crab, crayfish, and deer are used. The board has those six objects on it, also. We will play with bets being $1. The player places a bet on a number or object. The “house” rolls three dice. If none of the dice show the number or object that was bet, the house keeps the $1 bet. If one of the dice shows the number or object bet (and the other two do not show it), the player gets back his $1 bet, plus $1 profit. If two of the dice show the number or object bet (and the third die does not show it), the player gets back his $1 bet, plus $2 profit. If all three dice show the number or object bet, the player gets back his $1 bet, plus $3 profit.

Let $X$ = number of matches and $Y$ = profit per game.

• d List the values that $Y$ may take on. Then, construct one PDF table that includes both $X$ & $Y$ and their probabilities.
• e Calculate the average expected matches over the long run of playing this game for the player.
• f Calculate the average expected earnings over the long run of playing this game for the player.
• g Determine who has the advantage, the player or the house.

According to the South Carolina Department of Mental Health web site, for every 200 U.S. women, the average number who suffer from anorexia is one ( http://www.state.sc.us/dmh/anorexia/statistics.htm ). Out of a randomly chosen group of 600 U.S. women:

• d How many are expected to suffer from anorexia?
• e Find the probability that no one suffers from anorexia.
• f Find the probability that more than four suffer from anorexia.

The average number of children a Japanese woman has in her lifetime is 1.37. Suppose that one Japanese woman is randomly chosen.
( http://www.mhlw.go.jp/english/policy/children/children-childrearing/index.html MHLW’s Pamphlet)

• d Find the probability that she has no children.
• e Find the probability that she has fewer children than the Japanese average.
• f Find the probability that she has more children than the Japanese average.