Competition problem — Combinatorics

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Combinatorics Difficulty 5.0 AIME, harder Find the answer

## 1. On a Hunt

Three hunters shoot at a rabbit simultaneously. The chances of the first hunter succeeding are rated as 3 out of 5, the second hunter's chances are 3 out of 10, and finally, the third hunter's chances are only 1 out of 10. What is the probability that the rabbit will be killed?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

1. If the hare escapes from the hunters, it means that the first hunter missed (2 chances out of 5), as did the second (7 out of 10), and the third (9 out of 10). Therefore, the probability of a miss is $2 / 5 \cdot 7 / 10 \cdot 9 / 10=126 / 500$ . Consequently, the hare will be shot with a probability of $1-126 / 500=0.748$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.