3.2 Problem Set

1) In the carnival game Wiffle Roll, a player will roll a wiffle ball across some colored cups. Suppose that if the ball stops in a blue cup, the player wins $20. If it stops in a red cup, they win $10, and if it stops in a white cup, the player wins nothing. There are 25 white cups, 4 red cups, and 1 blue cup. Assume the chances of stopping in any cup is the same. How much should this game cost if it is to be a fair game?

2) In a simple game, you roll a single 6-sided die one time. The amount you are paid is the same as the amount rolled. For example, if you roll a one, you get paid $1. If you roll a two you get paid $2 and so on. The only exception to this is if you roll a 6 in which case you get paid $12. What should this game cost in order to be a fair game?

3) Suppose you are playing the game of GREED as described in Example 2. You have accumulated a total of 55 points on one turn so far. Is it to your advantage to roll one more time?

4) Suppose you are playing the game of GREED again. This time you have accumulated a total of 60 points in one turn so far. Is it to your advantage to roll one more time?

5) Using your results from numbers 4) and 5) and a little more investigation, for what number of points in a turn in the game of GREED does it make no difference if you roll one more time or stop? In other words, at what point total does the expected value with one more roll give the same total as if you had stopped?

6) In the Minnesota Daily 3 lottery, players are given a lottery ticket based upon 3 digits that they pick. If their 3 digits match the winning digits in the correct order, then the player wins $500. If the digits don't match, then the player loses. The game costs $1 to play. What is the expected value for a player of this lottery game.

[Figure10]

7) A bucket contains 12 blue, 10 red, and 8 yellow marbles. For $5, a player is allowed to randomly pick two marbles out of the bucket without replacement. If the colors of the two marbles match each other, the player wins $12. Otherwise the player wins nothing. What is the expected gain or loss for the player?

8) An insurance company insures an antique stamp collection worth $20,000 for an annual premium of $300. The insurance company collects $300 every year but only pays out the $20,000 if the collection is lost, damaged or stolen. Suppose the insurance company assesses the chance of the stamp collection being lost, stolen, or destroyed at 0.002. What is the expected annual profit for the insurance company?

9) A prospector purchases a parcel of land for $50,000 hoping that it contains significant amounts of natural gas. Based upon other parcels of land in the same area, there is a 20% chance that the land will be highly productive, a 70% chance that it will be somewhat productive, and a 10% chance that it will be completely unproductive. If it is determined that the land will be highly productive, the prospector will be able to sell the land for $130,000. If it is determined that the land is moderately productive, the prospector will be able to sell the land for $90,000. However, if the land is determined to be completely unproductive, the prospector will not be able to sell the land. Based upon the idea of expected value, did the prospector make a good investment?

10) A woman who is 35 years old purchases a term life insurance policy for an annual premium of $360. Based upon US government statistics, the probability that the woman will survive the year is .999057. Find the expected profit for the insurance company for this particular policy if it pays $250,000 upon the woman's death.

11) A bucket contains 1 gold, 3 silver, and 16 red marbles. A player randomly pulls one marble out of this bag. If they pull a gold marble, they get to pick one bill at random out of a money bag containing a $100 bill, five $20 bills, and fourteen $5 bills. If they pull a silver marble out of the bag, they get to pick one bill at random out of a bag containing a $100 bill, two $20 bills, and seventeen $2 bills. If your marble is red, you automatically lose. The game costs $5 to play.

[Figure11]

12) A spinner has four colors on it, red, blue, green, and yellow. Half of the spinner is red and the remaining half of the spinner is split evenly among the three remaining colors. A player pays some money to spin one time. If the spinner stops on red, the player receives $2. If it stops on blue, the player receives $4. If it stops on either green or yellow, the player wins $5. What should this game cost in order to be a fair game?

[Figure12]

13) A bag contains 1 gold, 3 silver, and 6 red marbles. A second bag contains a $20 bill, three $10 bills, and six $1 bills. A player pulls out one marble from the first bag. If it is gold, they get to pick two bills from the money bag (without replacement). If it is a silver marble, they get to pick one bill from the money bag, and if the marble is red, they lose. The game costs $3 to play. Should you play? Explain why or why not.

14) Suppose the Minnesota Daily 3 lottery adds a new prize. You still get $500 if you match all three digits in order, but you can also win $80 if you have the three correct digits but not in the right order. The game still costs $1. What is the expected value for a player of this lottery game?

Review Exercises

15) Consider the partially complete probability model given below.

[Figure13]

16) The student council is starting to prepare for prom and decides to name a committee of 6 members. Suppose that they decide the committee will have 2 juniors and 4 seniors on it. In how many ways can the committee be selected if there are 8 juniors and 8 seniors from which to select?

17) Three cards are dealt off the top of a well-shuffled standard deck of cards. What is the probability that all three cards will be the same color?":

18) A student does not have enough time to finish a multiple choice test so they must guess on the last two questions. List the sample space of the possible guesses for the last two questions if each question has only choices a, b, and c.

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