Puzzle time - bias test

[jon-nyc]

Before you are two coins; one is a fair coin, and the other is biased toward heads. You'd like to try to figure out which is which, and to do so you are permitted two flips. Should you flip each coin once, or one coin twice?

[Axtremus]

click to show

One coin twice.

[Klaus]

[Klaus]

click to show

I assume the solution doesn't depend on the amount of bias, so for simplicity let's assume that the bias is total and the biased coin will always land on its head.

With the "one coin flip each" strategy, we could identify the biased coin with 75% probability: If the unbiased coin yields T, we know for sure, and if it yields H we have to guess.

Now let's consider the "one coin twice" strategy.

With probability 5/8, we see HH. In this case, we guess that the coin is the biased coin. We'd be right 80% of the time.
With probability 3/8, we see at least one T. In this case, we guess that the coin is the unbiased one. We'd be right 100% of the time.

In total, we'd be right with probability 7/8, or 0.875, which is higher than the 0.75 from above.

The correct strategy is hence to flip one coin twice.

[jon-nyc]

Suppose you only get one flip.  If you flip coin A and it comes up heads, you will of course guess that it is the biased coin; if it comes up tails, you will guess that it's the fair coin. Now if you flip coin B and get the opposite face, you will be even happier with your previous decision.  If you get the same face, you will be reduced to no information and may as well stick with the same choice.  Thus, flipping the second coin is worthless.

Could flipping the same coin twice change your mind?  Yes.  If you get heads the first time, you are inclined to guess that that coin is biased, but seeing tails the next time will change your best guess to "unbiased."  We conclude that flipping one coin twice is strictly better than flipping each coin once.

As it turns out, it's a theorem that in trying to determine which is which of two known probability distributions, it's better to draw twice from one than once from each.  However, unlike in the puzzle above, it may happen that after seeing the result of your first draw you prefer to draw from the other distribution.  We'll leave it to the reader to concoct an example.

[The remarkable theorem mentioned above was proved by Romanian mathematician Gheorghe Zbaganu.]