Here’s a fun probability problem that Benoit.Jottreau@smartodds.co.uk showed me. If you’re clever at probability, you might be able to solve it exactly; otherwise it’s easy to simulate. But as with previous problems of this type, I think it’s more interesting to find out what you would guess the answer to be, without thinking about it too deeply.
So, suppose you’ve got 10 coins. They’re fair coins, in the sense that if you toss any of them, they’re equally likely to come up heads or tails. You toss all 10 coins. You then remove the ones that come up heads. The remaining ones – the ones that come up tails – you toss again in a second round. Again, you remove any that come up heads, and toss again the ones that come up tails in a third round. And so on. In each round, you remove the coins that come up heads, and toss again the coins that come up tails. You stop once all of the coins have been removed.
The question: on average, how many rounds of this game do you need before all of the coins have been removed?
There are different mathematical ways of approaching this problem, but I’m not really interested in those. I’m interested in how good we are, collectively, at using our instincts to guess the solution to a problem of this type. So, I’d really appreciate it if you’d send me your best guess.
Actually, let’s make it a little more interesting. Can you send me an answer to a second question as well?
Second question: same game as above, but starting with 100 coins. This time, on average, how many rounds do you need before all of the coins have been removed?
Please send your answers to me directly or via this survey form.
I’ll discuss the answers you (hopefully) send me, and the problems themselves in more detail, in a subsequent post.
Please don’t fill out the survey if you solved the problem either mathematically or by simulation, though if you’d like to send me your solutions in either of those cases, I’d be very happy to look at them and discuss them with you.