Partying with Bill Gates...and a Coin-Toss Quandary

The Puzzler

TOM: He's right, but he's gonna lose. Because it sounded very good. Maybe he just wanted Bill Gates to feel good. I mean, the guy's probably got an ego as big as Cleveland. Cleveland? As big as Australia, maybe.
RAY: Maybe.
TOM: And if the guy didn't fall for his little trick here, then Bill Gates might have lost a little of his self-esteem.
RAY: Well, anyway, Gates is gonna win his money.
TOM:: Yeah.
RAY: Unfortunately. Because it is going to happen, and it must happen.
TOM: Mmmm?
RAY: Let's say out of those 1,100 people we take, for the sake of convenience, 1,024 of them. It's just a number I picked out of the air.
TOM:: Just a number you picked out of the hat by doubling 512.
RAY: And let's take that 1,024 people and make 512 pairs of people.
TOM:: Yeah. OK.
RAY: OK? And each one of those pairs will do a coin toss. Well, obviously, one is going to . . . one of them is going to win.
TOM:: Mmm-hmm.
RAY: So, we're going to have 512 winners.
TOM:: Yeah.
RAY: OK? And conveniently, we take those 512 and we divide them into 256 pairs now.
TOM:: That's a nice number too.
RAY: Isn't it amazing?
TOM:: Yeah.
RAY: And, of course, we're going to have 256 people flipping coins. We're going to have . . .
TOM:: One hundred twenty-eight winners.
RAY: Son of a gun!
TOM:: Isn't that something?
RAY: And you do that nine times.
TOM:: Yeah.
RAY: And you're down to two people.
TOM:: Mmm-hmm.
RAY: And inevitably, one of them's going to win.
TOM:: Mmm-hmm.
RAY: So, of those people that played, you had two people that wound up doing it correct nine times, and one person who had to do it correct 10 times in a row.
TOM:: Right.
RAY: And Gates wins his money.
TOM:: And that was his bet.
RAY: He wins all the time, man! I'm getting sick of it!
TOM:: He's getting sick of it too.
RAY: Who's our winner?
TOM:: The winner is Mark Kime from Valparaiso, Indiana.

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