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## Another Game of Dice

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

 Puzzle ID: #43905 Fun: (2.51) Difficulty: (2.59) Category: Probability Submitted By: javaguru

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

"We each get one die, the highest die wins. If we tie, I win, but since you always lose when you roll a one, if you roll a one you can roll again. If you get a one the second time you have to keep it."

What is each person's probability of winning?

What are the probabilities of winning if you can keep rolling until you get something besides a one?

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 fame Jan 14, 2009 FIRST POST! tpg76 Jan 14, 2009 Another fun one - a bit easier than the previous, but fun. Good explanation as always. Thanks. MagaJamba Mar 15, 2009 I didnt understand a word you said sry opqpop Sep 27, 2010 Alternative way: Condition on when you roll a 1 or not. P(roll 1) = 1/6 P(win | you rolled 1) = (1 - 1/6) / 2 = 5/12 P(you don't roll 1) = 5/6 P(win | you don't roll 1) = P(win | you don't roll 1 and opponent rolls 1)P(opponent rolls 1) + P(win | you don't roll 1 and opponent doesn't roll 1)P(opponent doesn't roll 1) = 1 * 1/6 + ((1-1/5) / 2) * 5/6 = 1/2 Answer is 1/6 * 5/12 + 5/6 * 1/2 = 35 / 72. The second part was already answered above: 1/2. opqpop Sep 27, 2010 The (1-1/6)/2 and (1-1/5)/2 comes from symmetry. P(my # higher) = P(your # higher). P(my # higher) + P(your # higher) + P(tie) = 1. P(tie) is 1/6 and 1/5 for six and five sided die respectively. Stack1607 Jan 07, 2012 Speaking of the first game, I reached a different result: 121/216 the probability of my getting a higher die is 15/36 and the probability of a tie but above 1 is 5/36 - (2,2),(3,3),(4,4),(5,5),(6,6). Since I win in case of such tie, it should be counted, and the total of the previous probabilities is alone 20/36 or 40/72 - higher than the answer. The probability of a tie with both of us having 1 is 1/36. Multiply that by the probability of you getting another 1 on your second roll(1/6), since that's the only way i would win [i'm assuming that only you get to roll again while i keep my 1]. The result would be 1/216. Then the total probability of my winning is: 20/36 + 1/216 = 121/216 javaguru Jan 07, 2012 @Stack: You're mixing the rules up. You don't ever win a tie. Stack1607 Jan 16, 2012 It said "If we tie, I win, but since you always lose when you roll a one ...", which means, as I understood, in case of a tie other than (1,1), one person (i or you depending on the narrative) wins?! javaguru Jan 16, 2012 Yes, that's right. "I" win ties, "you" never win ties. eighsse Jul 28, 2013 Good one. I nailed it.

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