It may be overkill, but this setup screams binomial distribution in my opinion. What do you think?

@ChaseRyanTaylor Since the cups are selected without replacement, the hypergeometric distribution is used.

@N.F.Taussig Oh of course; good point

Possible duplicate of Probability of being poisoned

This reminds me a lot of the Monty Haul problem.

I bet she didn't really put poison in the 4th cup.

At the risk of going off-topic, remember "The dose makes the poison." If the cups of water are big enough, drinking three of them will kill you just as surely as the "poisoned" cup.

@aslum -- Is that the problem where you have to tow your prize from the door?

@wolfies She did.

How fast-acting is the poison? Do you mean drinking up to 3 of the 4 cups? If the poison is fast-acting and in the first cup chosen, then she won't have drunk out of 3 cups, but only 1.

@wolfies: the actual probability of her being poisoned is zero. She's spent the last several years building up an immunity. :-)

The downvoted answer of German R is correct; the rest are wrong. You have to be very careful when making conditional statements in probability problems. The answer to the question "if you pick three cups to drink, pour them together and drink them all at once, what is the probability of being poisoned?" is 3/4. But the problem: given that you drank the first cup, lived, drank the second cup, and lived, and got to the third cup, what is the chance you were poisoned on the third cup?" is 1/2, not 3/4. Assuming of course that the poison is instant death.

There are four cups. You get to choose one cup not to drink and if the cup you choose is not poison, What is the probability that you die?

@EricLippert German R is correct after the two cups, but the question was how likely it will be to get poisoned if she drinks 3 cups out of 4 (a priory) there is no statement that she will live to drink 3 cups OR that she immediatly dies after drinking. These are added assumptions not given by the problem. Just because she survived the first two in the realisation does not mean that the initial probability was 1/2 but 3/4. (which seems what the asker wanted to know)

If she knows she ca’t die, why such a complicated proof? Just a single immediate dose of poison would prove it (without a skeptic assuming she had a trick to avoid the bad one).

@KamiKaze: No one said the initial probability was 1/2. Under the fast-acting assumption, the probability of being poisoned, conditioned on the fact that three cups are imbibed, is 1/2. That's a conditional probability, not a standalone probability.

@BenVoigt But the question is about the initial probability as far as I can see. The conditional probabillites are just a bonus question and as there is no clear statement that the process ends after the poison was taken, even the 1/2 is debatable. Natural language is not a good medium for probability problems. Too many inaccuracies and possible interpretations make a clear formalisation hard.

@KamiKaze: I think we can agree that the question is confusingly worded. If the question was worded "If you attempt to drink up to three of the four cups, what is the probability of being poisoned on one of the attempts?" then plainly it is 3/4. But that's not the question that was asked.

@EricLippert: It's just an ambiguous question. It says "drink three of the cups" without specifying that they are mixed beforehand, but also not specifying they are imbibed in sequence. Could be drinking through three straws simultaneously.

Saw this on hot list question instantly thought: "Agents of Shield"