Betting on the Pennant
A friend proposes the following bet:
"Let's bet on three games the Giants will play. We have to guess the wins and losses in order, and whoever nails it wins."
"Hmmm," you say, "Since there's a 1/2 chance of correctly guessing a single outcome, that's a 1/8 chance of getting all three."
"Exactly. We'll say that ties don't count, and we just keep going until one of our sequences is a hit."
"That sounds like fun!"
"I'll let you name your sequence first, and I'll even give you 5:4 odds;
if you win I'll give you $50, but if I win you only need to pay $40."
"Seriously? No way I can pass up that bet!"
Is this a fair bet,
or is your friend taking advantage of you?
Your friend is taking you to the cleaners.
In this bet, making the second prediction gives you a huge advantage.
For
example, if you guess "win-win-win," your friend can guess
"lose-win-win," and you'll always lose unless the first three games are
wins.
Below are some first–second predictions of wins (W) and losses (L), and the probability that the second prediction hits first:
WWW -> LWW -> 7/8
WWL -> LWW -> 3/4
WLW -> WWL -> 2/3
WLL -> WWL -> 2/3
LWW -> LLW -> 2/3
LWL -> LLW -> 2/3
LLW -> WLL -> 3/4
LLL -> WLL -> 7/8