### Brain Teasers

# A Game of Dice

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

Your friend offers to play a game of dice with you. He explains the game to you.

"There are two dice and the goal is to have the highest total of two dice. You roll first. If you don't like your roll, then you can roll both dice again. If you roll a second time, you must keep the second roll. I'll roll second. I have to keep my first roll, but if we both roll the same number, I win."

On your first roll you roll a 7. Should you roll again or keep the 7?

Bonus question: if you play the optimal strategy, what is the probability that you will win?

"There are two dice and the goal is to have the highest total of two dice. You roll first. If you don't like your roll, then you can roll both dice again. If you roll a second time, you must keep the second roll. I'll roll second. I have to keep my first roll, but if we both roll the same number, I win."

On your first roll you roll a 7. Should you roll again or keep the 7?

Bonus question: if you play the optimal strategy, what is the probability that you will win?

### Answer

You should roll again.The probability of winning with a particular roll is the cumulative probability of your friend rolling one of the lower numbers.

There are 36 possible outcomes of rolling the dice (6 for the first die x 6 for the second die). The number of ways and probability of rolling each number is:

2,12: 1 = 1/36

3,11: 2 = 2/36

4,10: 3 = 3/36

5,9: 4 = 4/36

6,8: 5 = 5/36

7: 6 = 6/36

Since your friend wins ties, he wins with any number 7 or greater. There are 1+2+3+4+5 = 15 ways to roll a 2, 3, 4, 5, or 6. This gives you a

15/36 = .41666...

probability of winning if you keep the 7.

If you roll a second time, then you have to take the sum of the probabilities of each roll multiplied by the probability of winning with that roll. This gives:

2: 1/36*0 = 0

3: 2/36*1/36 = 2/1296

4: 3/36 * 3/36 = 9/1296

5: 4/36 * 6/36 = 24/1296

6: 5/36 * 10/36 = 50/1296

7: 6/36 * 15/36 = 90/1296

8: 5/36 * 21/36 = 105/1296

9: 4/36 * 26/36 = 104/1296

10: 3/36 * 30/36 = 90/1296

11: 2/36 * 33/36 = 66/1296

12: 1/36 * 35/36 = 35/1296

Adding these probabilities together gives a

575/1296 ~ .44367

probability of winning if you roll again.

Bonus question:

Since you now know that the optimal strategy is to reroll when you get a 7, and to keep anything 8 or higher, you can now calculate your chances of winning. Take the probability for each roll for the first throw of the dice multiplied by the probability of winning after that first roll. Any number 8 or higher has the same probabilities calculated above:

8: 5/36 * 21/36 = 105/1296

9: 4/36 * 26/36 = 104/1296

10: 3/36 * 30/36 = 90/1296

11: 2/36 * 33/36 = 66/1296

12: 1/36 * 35/36 = 35/1296

Adding these probabilities gives a 400/1296 ~ .31 probability of winning with the first roll.

Any number 7 or lower will be rerolled, and from above you know that the probability of winning on the second roll is 575/1296. The probability of rolling 7 or lower is 21/36 (from above: 1 - 15/36 = 21/36). This gives a probability of

2-7: 21/36 * 575/1296 = 12159/46656 ~ .26

And a

.31 + .26 = .57

probability of winning on the first or second roll.

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