Games of Chance and Expected Value
Expected value is a way of measuring what the long-term outcome of repeated trials of an experiment will be. This method is often used to determine the fairness of games of chance.
Nearly every game of chance has an expected value less than the cost of playing the game. If this were not true, these games would not be fund-raisers for governments, churches, and other charity groups. Casinos are also run in such a way to guarantee themselves profit. With this concept of expected value in mind, complete the following tasks and write up the results in a report.
Task 1. Consider rolling one (six-sided) fair die and tossing one (two-sided) fair coin. If the coin turns up heads, then the number of spots showing on the die is the score of that trial. If the coin turns up tails, then twice the number of spots showing on the die is the value for that trial. In other words, if the coin-toss results in a head and the die lands showing a 4 face up, then the score is 4 points for that trial. If the coin were to land tails up and the die landed a 4 face up, then the score of that trial will be 8 points. Determine the possible outcomes in the sample space and construct a probability distribution table. Once that is completed you are to find the expected value of the following experiment. If I were to invite you to play the above game where I would pay you your score in dollars (1 point = 1 dollar), what should I charge you to play? Using that ticket price, how many games would have to be played in order for me to have an expected profit of $100.00?
Task 2. Let’s consider the “same” experiment and scoring scheme in Task 1 except that now the die is “weighted” so that the probability of getting the various rolls of the die is as follows:
Roll 1 2 3 4 5 6
Probability 1/4 1/6 1/6 1/6 1/6 1/12
Find the expected value (don’t forget the coin toss). In order to make a little profit in the “long run”, what should your minimum charge be for the privilege of playing this game?
Task 3. The NC Education Lottery contains many games of chance. While it is certainly not the intent of this project to invite you to play the games of the lottery it does, nevertheless, provide some real world examples of probability outcomes and expected values. Go to the website http://www.nc-educationlottery.org/pick3.aspx and look at the rules for the “Carolina Pick 3”
Compute the expected value of one ticket purchased and compare that to the actual purchase price of the ticket.
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