Re:The Infamous Coin Toss
by petermcneeley on Wednesday August 02, 2023 @10:20PM
(#36980578)
Note a single lucky guy is not enough.
P of N heads is (1/2)^N but payout is (1.5)^N for this single lucky guy.
(The question can be reformulated such that this could be the case.)
Mmm, seems similar to the game of venture capital!
Wow, great point! I had ran my simulation several times to check that it was reasonably stable and it gave me less than $1 average wealth each time, so I thought that was that. But just now I re-ran it with a much larger population and indeed the average wealth grows now instead of shrinking. Sadly I can't edit my comment above any more, but:
> that the expected value of 1 iteration can be positive while the expected value of repeated iteration can be negative is very fascinating!
is indeed untrue. Thanks!
The intuition for why this happens is that the losses of the majority are made up by the big gains of a minority. You can even see that after two iterations:
Case 1: two heads -- 225% wealth Case 2: heads, tails -- 90% wealth Case 3: tails, heads -- 90% wealth Case 4: two tails -- 36% wealth
So, most people are losing money but a single lucky guy makes enough money that the expected value is still positive.
If you run a simulation with a low sample size, you might not see this effect because the "lucky guy" case is rare (and gets more rare with more iterations).