Question

A friend bets you $100 on a game involving two six-sided dice, one red and one green. You choose the number of times the pair of dice will be rolled. You win if the number of times a red 6 is rolled is at most 2 and the number of times a green 6 is rolled is at least 2. a) How many times should the dice be rolled to maximise your chance of winning? b) With that number of rolls, what are your expected winnings?

Answer 1

SOLUTION:

From given data,

A friend bets you $100 on a game involving two six-sided dice, one red and one green. You choose the number of times the pair of dice will be rolled. You win if the number of times a red 6 is rolled is at most 2 and the number of times a green 6 is rolled is at least 2.

Let X be the random variable no.of times 6 occurs on the red dice. Let Y be the random variable number of times 6 occurs on the given dice, then X Bin (n,1/6) and Y Bin (n,1/6) , n is same for both X and Y as the dice are rolled together,

Now , I win if X < 2 and Y > 2 and also note that X and Y are independent

Therefore,

P (X < 2 , Y > 2) = P (X < 2 ) P(Y > 2)

X (1 - P (Y < 1))

a) How many times should the dice be rolled to maximise your chance of winning?

No.of Trials	P (X < 2 )	P(Y > 2)	P(1 win)	P(1 lose )	Expected Win
2	1.000	0.028	0.028	0.972	-94.44
3	0.995	0.074	0.074	0.926	-85.25
4	0.984	0.132	0.130	0.870	-74.04
5	0.965	0.196	0.189	0.811	-62.14
6	0.938	0.263	0.247	0.753	-50.63
7	0.904	0.330	0.299	0.701	-40.28
8	0.865	0.395	0.342	0.658	-31.60
9	0.822	0.457	0.376	0.624	-24.84
10	0.775	0.515	0.400	0.600	-20.08
11	0.727	0.569	0.414	0.586	-17.25
12	0.677	0.619	0.419	0.581	-16.18
13	0.628	0.664	0.417	0.583	-16.65
14	0.579	0.704	0.408	0.592	-18.41
15	0.532	0.740	0.394	0.606	-21.19
16	0.487	0.773	0.376	0.624	-24.76
17	0.444	0.802	0.356	0.644	-28.89
18	0.403	0.827	0.333	0.667	-33.38
19	0.364	0.850	0.310	0.690	-38.08
20	0.329	0.870	0.286	0.741	-42.84

So the dices should be rolled 12 times.

b) With that number of rolls, what are your expected winnings?

Expected winnings = 100*P(I win) - 100*P(I lose)

For n=12

Expected winnings = -$16.18

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