One of the greatest hits of this blog was Elchanan Mossel’s dice problem, asked here and answered here. Here is how chatGPT (the free version) handles the problem.
You throw a die until you get 6. What is the expected number of throws (including the throw giving 6) conditioned on the event that all throws gave even numbers?
Image may be NSFW.
Clik here to view.
Image may be NSFW.
Clik here to view.
Small additions to the post:
1) Here are the results of a poll we had in 2017 about the answer
Image may be NSFW.
Clik here to view.
2. 3 is not the correct answer.
3. ChatGPT gets the right answer to the more basic question: You throw a die until you get 6. What is the expected number of throws (including the throw giving 6).
4. Luca Trevisan pointed out on FB a more extreme version of this problem (from a discussion on Reddit) where it defines
A= the expected number of rolls until you get the 100th 6, conditioned on rolling all evens, and
B = the expcted number of rolls until you get, for the first time, 100 6s in a row, conditioned on rolling all evens.
The claim was that both A and B are very close to 100, and that B<A, because of the way the conditioning disfavors long strings, and puts most of the probability on the shortest ways of achieving the condition.
(The naive incorrect computation will give you 300 for A and Image may be NSFW.
Clik here to view.
for B.)
5. For a discussion of the original question see on Math with bad drawing: A Probability Puzzle That You’ll Get Wrong ; On Mike’s math page: Exploring Elchanan Mossel’s fantastic probability problem with kids; ; Understanding The Math Behind Elchanan Mossel’s Dice Paradox; A recent discussion on Reddit; and quite a few other places…
Here is a very nice video by MindYourDecision
