The Greedy Pig

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Rules:

Greedy Pig is a one player game in which the objective is to accumulate as many points as possible without going bust. On each round the player rolls a pair of dice. If the roll does not include a two on either die, the players score is increased by the sum faces showing on the two dice. The player may continue to roll again and again as long as they score on each round. If either die has a two showing on the top face, the player loses all their points and ends the game with zero.

You may want to grab a pair of dice and try several games to experiment. If you prefer a calculator based version you can download a Ti-83+ program. You need appropriate computer software to connect to your calculator.

Several interesting probability questions can be asked. What is the best strategy to use to finish with the most points? What is the expected value a player will achieve using that strategy? Can you find the mean and standard deviation of scores after the first roll? Can you find the mean and standard deviation of the scores after n rolls for any n? Some of these are challenging problems, and you should feel confident of your probability talent if you can answer them all.

To help in exploring the game, here are three more Ti-83+ programs that might be of interest.
The first is a program that will run a game each time you press enter using a a strategy that continues to play as long as the score is below 18, and quits if the score is 18 or more. This has been suggested as a "Best" strategy. You can alter the limit to a different score (by changing the value in line 12 of the program) and try to get a better result.
The second will run fifty trials of the game and store the score and number of trials in Lists one and two on the calculator. You can alter both the number of rolls and the decision limit of 17 on this program.
The other program that may be of interest will play fifty games but each time the strategy is to stop after a certain number of rolls (controlled by an input), and again store results in L1 and L2. I found this important in testing the mean and standard deviation after n rolls.

I want to add some writing here about the computation of some of the questions asked above, but not yet... stay tuned, and have fun

In exploring these questions I also found use for a Fathom document that would allow me to run a thousand trials at a time. It is set up to show the total after two rolls and after three rolls, and display the results in a histogram and find the mean and standard deviation. Each row is one game and the columns show the results of each die rolled. Roll1 and roll2 for example are the first turn, etc. The "score" column shows the resulting score after the first two rolls (four dice) and the "score3" column shows the result after three turns.

Here is an alternate Fathom document that plays several rolls of the game. You can adjust the number for the strategy of playing another roll if the score is < N and stopping if the roll is > or = N.

Some discussion of the results It is easy to show that the probability distribution for the first roll results looks like this:
Probabilities given in 36ths
Score 0___2__4__5__6__7__8__9__10__11__12
Prob 11__1__2__2__3__4__3__4___3___2___1

From this we get a mean score after one round of 95/18 or about 5.27 points with a standard deviation of about 4.04 points. We can also see that the average each round for the people who do not "go bust" is 38/5 with a standard deviation of about 2.433. This can make it easier to calculate the expected value after N rolls for people playing under the "Pure greed" strategy of play till you bust.

For those in the sub-population who have not gone bust, the score after n rounds is the sum of n populations with a mean of 38/5 so we can use what we know about combining distributions to find the expected value of this lucky subpopulation. Since all the other players are at a score of zero, and since we can easily compute what percentage have gone bust after any number of rounds, then we can combine these using the weighted percentages of each population to find the expected value of the total population after N rolls.

Here are some computations for the first few rolls (which I hope is correct)
Rolls________perc gone bust______perc surviving______Mean_survivors______Ex Value
1____________11/36______________25/36_______________ 38/5 ____________ 38/5 * 25/36
2___________1-(25/36)2_________(25/36)2____________76/5________(76/5)(25/36)2
but a pattern emerges that seems to suggest that the Expected value after any number of turns is going to be E(n)= n(38/5)(25/36)n

Rolls_n _____1______2______3_____4______5
E(n)________5.27___7.33___7.635_7.07___6.137

These values seems to agree with simulated results using the Ti-83+ program pigrolls. The maximum expected value occurs after three rolls, then decreases to zero as n goes to infinity.

Some comments on calculating the standard deviation after n rolls.... yet to come