I don't care to belong to a club that accepts people like me as members. --Groucho Marx
Let's play the following game: I choose a random fraction of the population from 0 to 100%. This choice is completely unbiased and uniform. Whatever that number is, I choose that fraction of the population, again at random, and mail them all little cards saying, "welcome to the RED TEAM." The remaining (100-P)% of the population gets little cards saying, "welcome to the BLUE TEAM."
You get one of my cards in the mail. You have no information about what choices I made, and you haven't even opened the envelope yet.
Before you look: what is the likelihood that your team is at least half the population? Take a guess and click for the answer.
The answer is 3/4. There is a 3/4 probability that your team, whatever it is, is larger than half.
You may think the probability is 1/2, because I chose a random fraction of expected value 50%. The expected size of the RED team is 50%, and the expected size of the BLUE team is 50%. The expected size of either team is 50%. All true, but the expected size of your team is 2/3. Why? Because you're on it.
This is an example of the Groucho Marx paradox: if I choose a random group with a random size, the expected size of that group changes, sometimes dramatically, depending on whether or not you are a member.
The mathematics behind this derives from conditional probability. Let 0<=P<=1 be the fraction of the population comprising the RED team. I chose P uniformly between 0 and 1, so its probability distribution is flat.</p>
Now let's look at you: if you know my choice was P=p, what is the probability that you will be picked for the RED team? It's just p:
Pr[ RED | P=p ] = p
Pr[ BLUE | P=p ] = (1-p)
The important fact here is that when P is large, you're more likely to be a member of the RED team. But conditioning works both ways: if you're a member of the RED, it's more likely that P was large. We can see this by reversing the direction of conditioning. This can be achieved using Bayes Rule, or we can just figure it by inspection:
Pr[ P=p | RED ] = (Pr[RED and P=p]/Pr[RED])
= (Pr[RED|P=p]Pr[P=p]/Pr[RED]) = pdp/Pr[RED] = 2p dp
Pr[RED] is the probability that any random person is assigned RED, in the absence of any other information. By symmetry, this is 1/2.
In semi-English: before you know your membership, P has a uniform distribution over all possible values. If you discover that you are on the RED team, P's distribution becomes a ramp with 100% on the high end:
Likewise, if you discover that you are BLUE, P suddenly has a ramp distribution from 100% down to 0:
In either case, you can compute the expected value:
E[ P | RED] = ∫ x⋅2xdx = 2/3,
E[ P | BLUE] = ∫ x⋅(2-2x)dx = 1/3,
PR[ P≥1/2 | RED] = ∫ 2xdx from 1/2 to 1 = 3/4
PR[ P≥1/2 | BLUE] = ∫ (2-2x)dx from 1/2 to 1 = 1/4
In either case, the expected size of the team you're on is always 2/3, and the probability that your team has size ≥1/2 is always 3/4.
Summary
This is a "paradox," which is not a logical contradiction or impossibility. A paradox is a perfectly legitimate mathematical result that just happens to violate our intuition. Conditional probability is pretty counterintuitive stuff: many paradoxes, including the Monty Hall paradox, are essentially paradoxes of conditioning.
If you're confused, just think of it this way: "conditioning" is just another name for revising the odds when you gain new information. Here, the new information is that you witnessed an event---your selection for a group---that is highly correlated with the thing you're trying to guess.
In the end, Groucho Marx was right. You probably don't want to belong to a club that chooses you as a member, because according to the math, it isn't all that selective.