OC just for you! ♥️
Most people would fail to understand the question. So many will flip the switch a bunch of times randomly. In other words: This would be super frustrating for the villain.
There surely must’ve been a more comprehending way to phrase the dilemma.
Info: How many people live in the kingdom?
There’s a five in six chance you are picked 1 out of X, and a one in six chance you are 10 out of X.
If you’ve been picked, there are three possible outcomes.
Flipping the lever kills you. 5/6 x 1/X
Flilling the lever saves you and 9 other people. 1/6 x 1/X
Flipping the lever does nothing at all. 1/6 x 9/X
From a purely statistical standpoint, you’re five times more likely to die flipping the lever, but the expected value, measured in lives saved, for flipping the lever is twice as high as not.
From a purely altruistic measure, you should always flip the lever, because at worst you kill yourself, at best you save 10 people, and you can do it with significant confidence that it doesn’t actually matter.
But back to my original question, 5/6X vs 1/6X vs 9/6X where as X approaches infinity, the difference becomes negligible.
Good question to ask, since specifics of selection process may affect the decision outcome! Other variants include growing humans in a vat from scratch on demand, using Star Trek transporter clones, or abducting the necessary number of people from a pre-selected list where your name happens to be the first one. For now, imagine the potential population as the 5 billion living cognizant adults.
as X approaches infinity, the difference becomes negligible
It may be negligible to the 4.999… billion adults sleeping comfortably and securely in their beds tonight, but the problem presupposed that you have already been abducted. It remains underdefined whether you refers to you the specific person reading this meme, or a more general you-the-unfortunate who has been chosen and is now listening to this on the headphones.
Noise cancelling earphones sucks at blocking voices. Just yell and ask if there are others.
Or spit, or blow air at your potential neighbor, or fart in their general direction!
That’s assuming the villain who is trying to deny you information by the blindfold and earplugs was dumb enough to put them close together that a spit would reach a neighbor.
tldr: Always flip the switch
Edited with some of TauZero’s suggested changes.
- 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
- 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.
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.
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.
No matter what your goal is, given the information you have, flipping the switch is always the better choice.
Excellent excellent!
If 6 is rolled, then P(X|R=6) = (N-1 choose 9)/(N choose 10)
Might as well reduce that to 10/N to make the rest of the lines easier to read.
If you don’t flip it, you have a 2/3 chance of dying.
There is also a chance that your switch is not connected and someone else has control of the real one. So there is an implicit assumption that everyone else is equally logical as you and equally selfish/altruistic as you, such that whatever logic you use to arrive at a decision, they must have arrived at the same decision.
No matter what your goal is, given the information you have, flipping the switch is always the better choice.
That is my conclusion too! I was surprised to learn though in the comment thread with @pancake that the decision may be different depending on the percentage of altruism in the population. E.g. if you are the only selfish one in an altruistic society, you’d benefit from deliberately not flipping the switch. Being a selfish one in a selfish society reduces to the prisoner’s dilemma.
There is also a chance that your switch is not connected and someone else has control of the real one. So there is an implicit assumption that everyone else is equally logical as you and equally selfish/altruistic as you, such that whatever logic you use to arrive at a decision, they must have arrived at the same decision.
Ah, yes. I forgot to account for that in my calculations. I’ll maybe rework it when I find time tomorrow.
