Puzzle time - peek advantage

[jon-nyc]

You are about to bet on the color of the top card of a well-shuffled deck of cards (of which 26 are red, and 26 black).  You get to pick the color to bet on; if you're right you win $100, otherwise you lose the same.

How much is it worth to you to get a peek at the bottom card of the deck?  How much more for a peek at the bottom two cards?

[Klaus]

The first question sounds pretty simple.

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98 Cents.

[jon-nyc]

I get twice that for the first.

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100/51

[Klaus]

Hm, am I missing something?

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The chance of winning should be 26:51, which means on average you'd win 50.98$. Without peeking, the average would be 50$. Hence the .98$

[jon-nyc]

There’s also losing to take into account.

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With one card known your expected payoff is

(+100)(26/51) + (-100)(25/51)
= 100/51

[Klaus]

Oh, I completely missed the "otherwise you lose the same" part.

[jon-nyc]

Second half is surprising.

[Horace]

Slightly more than half the time, two card peeks give no advantage. Slightly less than half the time you get the advantage of seeing two of the same color.

[jon-nyc]

Well sure, but how’s it net out?

[Horace]

Intuitively it is indistinguishable from the one card peek.