Problem 26 · 2024 Math Kangaroo
Stretch
Counting & Probability
casework
Sylvia has several fair 12-sided dice, each with the numbers 1 to 12 written on their faces. If she rolls all the dice simultaneously, the probability of rolling exactly one 12 is equal to the probability of not rolling a 12 at all. How many dice does Sylvia have?
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Answer: D — 11
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Hint 1 of 2
Write the probability of exactly one 12 and of no 12 for n dice, then set them equal.
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Hint 2 of 2
n·(1/12)(11/12)^(n−1) = (11/12)^n simplifies to a single equation for n.
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Approach: set the two probabilities equal
- P(no 12) = (11/12)ⁿ and P(exactly one 12) = n·(1/12)·(11/12)^(n−1).
- Setting them equal and dividing by (11/12)^(n−1): n/12 = 11/12.
- So n = 11 dice.
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