Problem 10 · 2011 Math Kangaroo
Medium
Counting & Probability
careful-counting
Max and Hugo roll a number of dice to decide who jumps first into the cold lake. If no six comes up, Max jumps; if exactly one six comes up, Hugo jumps; if several sixes come up, neither jumps. How many dice must they use so that the two boys are equally likely to have to jump in?
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Answer: B — 5
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Hint 1 of 2
Write the chance of 'no six' and the chance of 'exactly one six' with n dice, then set them equal.
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Hint 2 of 2
(5/6)n = n·(1/6)(5/6)n−1 collapses nicely.
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Approach: set the two probabilities equal and simplify
- P(no six) = (5/6)n; P(one six) = n·(1/6)·(5/6)n−1.
- Setting them equal and dividing by (5/6)n−1 gives 5/6 = n/6.
- So n = 5.
Mark:
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