Aha!

What Hacker News says

1. supernova87aJan 2022
   Watching the trailer, makes me think that people who may enjoy this series will also enjoy a book called "Aha" by Martin Gardner (from probably 40 years ago)[0].

   It has a lot of amusingly illustrated insight problems that show you how thinking about a problem differently simplifies getting to the answer a lot. It's probably aimed at advanced teenagers, but in fact the lessons there are some that even many adults probably have not come across before. I mean, I haven't watched the Masterclass, but I imagine Terrence Tao isn't teaching multimanifold group theory or something anyway, so this is the kind of insightful more daily-math problems he's helping people to understand how to think about.

   [0] search your favorite, ahem, LIBrary GENeric source of books to find it.

   Show morethread
2. nwhiteheadJul 2011
   Simpson's paradox is amazing, it's worth reading all the examples in the article and really understanding it.

   Another good source is Martin Gardner's "Aha! Gotcha" book. He presents the paradox as a woman trying to find eligible bachelors at a party. In room 1 her odds are better if she goes for guys with mustaches. In room 2 her odds are also better if she goes for guys with mustaches. But when everyone goes into one room, her odds are better for guys without mustaches. It's incredible.

   Show morethread
3. robocatMay 2022
   > Having chosen an envelope at will, but before inspecting it, you are given the chance to switch envelopes. Should you switch?

   Normally we would assume there is someone who owns the envelopes of money. Now their incentive could be to offer to switch only if the envelope you chose contains more money (so they have a chance of keeping more). More like Monty Hall.

   Even without that, after switching, should you switch back? The equation given argues you should switch back, so obviously the equation must be wrong.

   It is easier to think: if I switch I have a 50% chance of making $X and a 50% chance of losing $X. Switching doesn’t change expected $.

   I actually think the later problem in the Wikipedia article is more clever:

     Two people, equally rich, meet to compare the contents of their wallets. Each is ignorant of the contents of the two wallets. The game is as follows: whoever has the least money receives the contents of the wallet of the other (in the case where the amounts are equal, nothing happens). One of the two men can reason: "I have the amount A in my wallet. That's the maximum that I could lose. If I win (probability 0.5), the amount that I'll have in my possession at the end of the game will be more than 2A. Therefore the game is favourable to me." The other man can reason in exactly the same way. In fact, by symmetry, the game is fair. Where is the mistake in the reasoning of each man? — "Martin Gardner: Aha! Gotcha"

   Show morethread