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.
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.
I am always surprised how my first guess gets wrecked by Bayes rule. I would have thought that there is 5/6 chance I am on side track and 1/6 that I am on the main track.
I am confused as to why anyone would not flip the switch? Flipping the switch seems to have somewhere between a 10% and 100% chance of saving your life, and not flipping the switch seems to guarantee death?
Is there some kind of penalty to flipping the switch that I am missing?
Or is the drawing misleading, and in Scenario B, there is also supposed to be a person drawn on the other track?
You are on the side track in scenario A. You die if you pull. Ironically, you’d be killing yourself. The dice are to make the two scenarios not equally likely.
there’s no way to know which track the trolley is on
It’s a standard trolley meme problem, the trolley will keep going on the main track unless the lever is switched 😁. I thought !science_memes would be familiar with trolley problems, but I guess I get to introduce some of you! You might want to start off on some easier trolley memes first, this is advanced level stuff.
where the real lever sends it
There is not usually ambiguity with the lever. If you wish, you can have an announcement in the headphones “main track… side track…” every time you flip the lever. Your only uncertainty is which track you yourself are bound to, given how you’re blindfolded.
Yeah, if I woke up tied to train tracks and had someone explain that to me, I’d zone out and then panic because I had no idea what the fuck was going on
I don’t get it. Do we know that the trolley is heading for the people or not? Do we know if flipping the switch moves it away from whatever track that the people are on? Or is it going in the main track in all instances unless you hit the switch?
I assume a villain would aim the trolley at the people, regardless of what track they’re on. That’s why they’re villains. So I would always flip it.
It’s a standard runaway trolley problem. The trolley is traveling down the main track unless the switch is flipped to send it down the side track. The lever is labeled such that there is no ambiguity which way it is set, the blindfolds notwithstanding. The villain is pernicious and will be equally (though not exceedingly so) delighted to see you die by your own action where inaction would have had saved you. You can somehow trust that the announcement in the headphones is true and not a lie. Such as, for example, you have seen this exact situation happen many times before on TV and survivors/witnesses have described the villain to be truthful every time.
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.
From a purely utilitarianism perspective, assuming all utility is linear and unscaled:
5/6 chance I’m on the side track * 1 person saved = 5/6
1/6 chance on the main track * 1/10 chance my switch is real * 10 people saved = 1/6
Seems pretty clear that you should not flip the switch. However, if I am on the main track, this thinking will lead to no-one flipping the switch and no lives saved whereas everyone thinking it will lead to a guaranteed save -> utility of 10/6.
If I can assume more than half the people can be rational and will think like me then I should flip the switch.
Except that if people are chosen randomly there is 2/3 chance that you are on the main track according to Bayes. Let’s assume there are 10 people.
The probability to be chosen is 1/6 (all are chosen if 6 is rolled) + (5/6) × (1/10) (only one is chosen to go to the side track if 1-5 is rolled) = 15/60 = 1/4.
The probability that you are on the side track knowing that you have been chosen is the probability that you have been chosen knowing that the side track is selected (1/10) × the probability that the side track is selected (5/6) divided by the probability for you to be selected at all (1/4), so (1/10)×(5/6)/(1/4) = 20/60 = 1/3. So there is a 2/3 chance that you are on the main track.
If you do not flip the switch, (2/3)×10 = 20/3 people die.
If you flip the switch, 1/3 (you if on side track) + 10 × 2/3 × 9 / 10 (switch misfires 9 out of 10 times if on the main track) = 190/30 = 19/3 die. This is slightly better than not flipping the switch, you save 1/3 people more. That’s an arm and a leg.
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.
I love the idea of the villain explaining the whole thing to the captive and at the end being like, “okay, I’m about to put the blindfold and noise canceling headphones on, so this is the last chance for any other questions about how this works”
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.
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