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Dec. 22, 2004
A Dicy Bet
A cube has six faces. A die is a cube that has six faces marked with from one to six dots.
Alvin bets that if a die is thrown four times in a row, it's bound to show 1 exactly once.
Betty bets that, in four throws, 1 will either not appear at all or show up more than once.
Who has the better chance of winning?
MathGolf: Challenge 14
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The chance of 1 coming up in one throw is 1 in 6.
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Betty has a better chance of winning the bet.
The number of possible outcomes after four throws of the die is 6 x 6 x 6 x 6 = 1,296.
Suppose that the die has already been thrown once and a 1 appeared. For the remaining three throws, the number of possible outcomes that doesn't show a 1 is 5 x 5 x 5 = 125. In the same way, if 1 appeared on the second, third, or fourth throw and no others, the number of possible outcomes for each case is 125. Altogether, there are 125 + 125 + 125 + 125 = 500 outcomes that favor Alvin.
The number of outcomes in which 1 does not appear at all or appears more than once is 1,296 – 500 = 796. Because more possible outcomes favor Betty, she's more likely to win the bet.
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