Two Coins Revisited

Last time we looked at the classic two coins problem, which (as a reminder) goes like this:

The host flips two coins. If both are heads, he flips again; if at least one is tails, he announces this fact. The contestant has to guess whether the other coin is heads or tails.

Without getting into the details, the contestant has a 2/3 chance of guessing correctly if he picks heads. The probability argument is simple, but nobody believes it anyways, so I won't repeat it here.

Clearly this is exploitable, since we can offer just above 50/50 odds to our mark as long as we can get them to play the host.

What I'm thinking about now: is it possible to very slightly alter the game so that we could let the mark play the contestant, at the same odds, and still make money? In other words, what is the minimal modification that actually turns this into a 50/50 game?

My proposal is that instead of canceling the game and re-flipping if we hit two heads, the host now gets to pick whether he tells the contestant about a head, or a tail, and we play each round.

Huh? WTF am I on, right? Isn't the game completely symmetrical, and we're just flipping it around so that we don't have to discard any tosses?

Au contraire. Let's enumerate the possibilities, assuming that if HT or TH comes up my choice of which to announce is completely random:

• (25%) HH - announce heads
  
• (12.5%) TH - announce heads
  
• (12.5%) TH - announce tails
  
• (12.5%) HT - announce heads
  
• (12.5%) HT - announce tails
  
• (25%) TT - announce tails
  

No big surprises there.

But let's start our arithmetic. Suppose I've announced heads. Out of a total 50% probability for that case, 25% comes from HH, 12.5% comes from TH, and 12.5% comes from HT. In other words, given that I've announced heads, there's exactly a 50/50 shot that the other coin is heads!

So what happened to the 2/3 from the correct answer to the previous problem? It's gone, plain and simple, and we're back to a fair even gamble again. You can figure out where it went by looking at that table above: now instead of being forced to announce heads in both the TH and HT case, we can split our call 50/50, and the bias is gone.

If you ever try this in real life, the way to do it is to first take the mark's money for a while as the contestant, and then when they're starting to get the sense that they've been had, suggest that you for expediency's sake you just start playing every flip, and that they can choose whether to call heads or tails when they flip. Only let this go for a round or two, though, because you're now playing a slightly losing bet. Soon thereafter, suggest that maybe they'd have better luck on the other side of the table, and let them play the contestant. Pull it off right, and they'll never even suspect that you slipped in a rule change that tilted the game in your favor; as long as you then proceed to play longer as the host under the new rules than you did as the contestant, the ruse has decidedly positive expected value, and the additional bonus that you leave the mark feeling as if they had a fair shot the whole time.

I'd wager some pretty good bank that you could even sucker quite a few people that understand the original problem into this version...