Poisoned vials
Seven indistinguishable vials are placed in a row. A random vial is poisoned. The vials were “swapped” 3 times, where a “swap” means switching the positions of two vials. At the end of this process, the probability that the leftmost vial is poisoned can be written as , where and are relatively prime integers. Find .
The answer is 8.
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1 solution
The position of the vials is unimportant - the probability that the leftmost vial is poisoned at the end is equal to the probability that any other vial is poisoned at the end.
Since there are 7 vials in total, the probability that the leftmost one is poisoned is simply 7 1 .