A Collaboration between: Los Angeles USD University of California, San Diego San Diego State University University of California, Irvine Preparing for Success in Algebra English Language Learners in Mathematics

Chuck-a-Luck Problem Scenario: The game of Chuck-a- Luck is an old carnival game which we wish to analyze in this activity. You’ll have an opportunity to simulate the game and delve into its mathematics.

The “player” simply chooses a number from 1 to 6 and bets on it. Three dice are rolled and if the chosen number appears on one die, the player gets his/her money back plus the same amount. If two dice show the chosen number, then the player gets his/her original money back plus twice the amount. Finally, if all three dice show the chosen number then the player gets his/her original amount plus three times the amount.

For example, if I bet $1 on the number 5, and after the roll of the three dice two 5’s show heads-up, I get my $1 back plus $2 more. It is pretty simple to play and to simulate in a classroom. Here are your tasks.

Procedure Please rearrange yourselves at your tables as necessary so there are 8 people at each table and create partners so there can be 4 simultaneous instances of the game at each table.

Select one person in the partnership to be the carnival employee who rolls the dice and the other the carnival attendee who selects the number and wagers $1 on each roll.

Play the game with your partner 25 times with the carnival employee keeping track of the attendee's wins and losses on the accompanying table.

Create a total of 100 instances by combining the results of all 4 groups at your table. If the player is ahead by $7 for example, express that as +7. If the carnival is ahead by $7, express that as -7.

Combining the Results We will then combine the results of all the tables which will simulate several hundred instances of the game. We would expect the results from so many instances to be close to the theoretical results which we would like to analyze.

InstanceWin 1$Win 2$Win 3$Lose 1$ ……….…………..……………. …………… Totalstimes 1times 2times 3times -1

Sample Space The sample space is an exhaustive list of all the possible outcomes of an experiment. Each possible result of such a study is represented by one and only one point in the sample space. For ease of calculation, we have three dice of different colors so they can be distinguished. What is the sample space for this experiment?

An Event An event E is any collection of outcomes of an experiment. Formally, any subset of the sample space can represent an event.. For example if we are considering the event E of obtaining exactly one 5, then the number of outcomes that have exactly one 5 form the event. The possible number of 5's for any outcome are 0,1,2,or 3.

Independent Events Independent events are events such that the outcome of one event has no bearing on the outcome of the other event e.g. if we roll two dice, the value on one of them is unaffected by the value on the other.

Probability Probability provides a quantitative description of the likely occurrence of a particular event. Probability is conventionally expressed on a scale from 0 to 1; a rare event has a probability close to 0, a very common event has a probability close to 1.

Equally likely outcomes In some experiments, all outcomes are equally likely. In our sample space above, not all outcomes are equally likely. In a raffle, all raffle ticket holders are equally likely to win, that is, they have the same probability of their ticket being chosen.

Mutually Exclusive Two events are mutually exclusive if both cannot occur simultaneously. Getting an odd or an even number on the roll of a die are mutually exclusive events. Getting an odd number and getting a number greater than 3 are not. We add probabilities to get the probability of one mutually exclusive event or the other.

An Example For example, if one event is obtaining a 1 or a 2 on one roll of a die, the probability of this event is 1/3. A second event could be getting a 5 or 6 which also has probability 1/3. The probability of getting one or the other of these mutually exclusive events is 1/3+1/3 = 2/3.

Independent Events We obtain the probability of two independent events both occurring by multiplying their respective probabilities. For example, if we roll two dice, the probability of obtaining a one on both dice is calculated as 1/6 times 1/6 or 1/36.

Expected Value The expected value, EV is the result that should occur if we perform the experiment for an extended period of time but it is usually expressed in terms of a single instance which may or may not be possible to achieve.

The Expected Value for Chuck-a-Luck EV = -$1 (125/216) + $1 (75/216) + $2 (15/216) + $3 (1/216) = -$(17/216) ≈ -$0.08 On average, you should expect to lose approximately $0.08 each time you play a round of the game. In 100 games, one should expect to lose $8.

A Fair Game A fair game is a game where the expected value is 0 i.e. you expect to be even after an extended period of time playing the game. The Chuck-a-Luck Game is not fair as we have seen.

Making it Fair How might the rules of the game be changed so that it is fair? Can you see how to do it by adjusting the payouts for two 5's and/or three 5's?