# Brain-aching list of counterintuitive probability riddles

In 2017, Leila asked Stack Exchange for suggestions for counterintuitive probability riddles for a course on probability; the assembled list is a brain-aching adventure in Monty Hall problems, neighbors' daughters, Sleeping Beauty epistemology, colored lottery balls and birthday paradoxes.

A variation on Penny's game: A new family moves to your block. Since you have children you are interested in what children they have: (1) You learn that they have two children, one of which is a girl. What is the probability the other child is also a girl? (2) You learn that they have two children, the older child a girl. What is the probability the other child is also a girl? Tell why the answers are NOT the same. – user247327 Feb 16 '17 at 16:45
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@user247327 "Tell why the answers are the SAME" is also valid here. As it all depends on the assumptions you make for this question. The way it is phrased in this comment I would say 1/2 in both is the correct answer. The problem is that you assume a distribution with 4 states. Which is just an assumption for the stated problem. en.wikipedia.org/wiki/Boy_or_Girl_paradox – Kami Kaze Feb 17 '17 at 10:31
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Mildly interestingly, the random walk problem you mentioned in higher dimensions is more complicated. For n=2, the same result applies: one returns to the origin a.s. In higher dimensions, the probability of returning to the origin is strictly between 0 and 1. – anomaly Feb 18 '17 at 4:38

Counterintuitive examples in probability
[Leila/Stack Exchange Mathematics]