I'll give you a dollar if it comes up 0,1,2,6,7, or 8. You give me a dollar if it comes up 3,4, or 5. You have twice as many outcomes as me, so you should make out like a bandit!

Q: Why would I play flip-8?

What are the odds of heads coming up 0 times. This is n-choose-m again. 8 choose 0 is 1. There are 256 ways to flip 8 coins, and 1 way to get a zero.

To get 1, it's 8 choose 1 = 8.

To get 2, it's 8 choose 2 = 28.

To get 6 it's 28. To get 7 it's 8. To get 8 it's 1.

It all adds up to 1+8+28+28+8+1 = 74. So the expected payoff for you is 74/256, or 0.2890625.

Since 8 choose 3 is 56, and 8 choose 4 is 70, my chances of winning are (56+70+56)/256 = 182/256 = 0.7109375.

Seems like I could make a lot of money of this game if I could convince someone to play...

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