howrar, 2 years ago (edited 2 years ago)

tldr: Always flip the switch

Edited with some of TauZero’s suggested changes.

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• Let N be the size of the population that the villain abducts from
• Let X be the event that you are abducted
• Let R be the outcome of the villain’s roll
• Let C be the event that you have control of the real switch

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• If 1-5 is rolled, then the probability that you are abducted is P(X|R∈{1,2,3,4,5}) = 1/N
• If 6 is rolled, then P(X|R=6) = (N-1 choose 9)/(N choose 10) = ((N-1)!/(9! * (N-10)!)) / (N!/(10! * (N-10)!)) = 10/N
• The probability of getting abducted at all is P(X) = P(X|R∈{1,2,3,4,5})P(R∈{1,2,3,4,5}) + P(X|R=6)P(R=6) = (1/N)(5/6) + (10/N)(1/6)
• The probability that a six was rolled given that you were abducted: P(R=6|X) = P(X|R=6)P(R=6)/P(X) = (10/N)(1/6)/((1/N)(5/6) + (10/N)*(1/6)) = 2/3

So as it turns out, the total population is irrelevant. If you get abducted, the probability that the villain rolled a 6 is 2/3, and the probability of rolling anything else is its complement, so 1/3.

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Let’s say you want to maximize your chances of survival. We’ll only consider the scenario where you have control of the tracks.

• P(C|R∈{1,2,3,4,5}) = 1/10
• P(C|R=6) = 1
• P(C) = P(C|R∈{1,2,3,4,5})P(R∈{1,2,3,4,5}) + P(C|R=6)P(R=6) = (1/10)(5/6) + (1)(1/6) = 1/4
• P(R=6|C) = P(C|R=6)P(R=6)/P(C) = (1)(1/6)/(1/4) = 2/3
• P(R∈{1,2,3,4,5}|C) = P(C|R∈{1,2,3,4,5})P(R∈{1,2,3,4,5})/P(C) = (1/10)(5/6)/(1/4) = 1/3
• If you flip the switch, you have a 1/3 chance of dying.
• If you don’t flip it, you have a 2/3 chance of dying.

If you want to maximize your own probability of survival, you flip the switch.

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As for expected number of deaths, assuming you have control of the tracks:

• If you flip the switch, the expected number of deaths is (1/3)*1+(2/3)*0 = 0.33.
• If you don’t flip it, the expected number of deaths is (1/3)*0+(2/3)*10=6.67.

So to minimize the expected number of casualties, you still want to flip the switch.

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No matter what your goal is, given the information you have, flipping the switch is always the better choice.