5.13 Discrete and continuous random variables: homework  (Page 3/5)

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.

Consider the following experiment. You are one of 100 people enlisted to take part in a study to determine the percent of nurses in America with an R.N. (registered nurse) degree. You ask nurses if they have an R.N. degree. The nurses answer “yes” or “no.” You then calculate the percentage of nurses with an R.N. degree. You give that percentage to your supervisor.

• a What part of the experiment will yield discrete data?
• b What part of the experiment will yield continuous data?

When age is rounded to the nearest year, do the data stay continuous, or do they become discrete? Why?

Births are approximately uniformly distributed between the 52 weeks of the year. They can be said to follow a Uniform Distribution from 1 – 53 (spread of 52 weeks).

• a $X~$
• b Graph the probability distribution.
• c $f(x)=$
• d $\mu =$
• e $\sigma =$
• f Find the probability that a person is born at the exact moment week 19 starts. That is, find $P(x=\text{19})=$
• g
• h Find the probability that a person is born after week 40.
• i $$
• j Find the 70th percentile.
• k Find the minimum for the upper quarter.