# Three Choices with One Coin Puzzle - Solution

### The Puzzle:

At a restaurant, how could you choose one out of three desserts with equal probability with the help of a coin?

Bonus: What if the coin is biased and the bias is unknown?

Bonus: What if the coin is biased and the bias is unknown?

### Our Solution:

Toss the coin twice.

Let TH, HT and TT correspond to the three choices.

And if you get HH, just repeat (so it takes 8/3 tosses on average).

BIASED COIN

If the coin was biased, TH and HT would occur with equal probability.

So you could assign THHT, HTTH and THTH to the three choices, with other 4-toss outcomes rejected.

Or you could assign HTT, THT and TTH to the three choices, with other 3-toss outcomes rejected.

Let TH, HT and TT correspond to the three choices.

And if you get HH, just repeat (so it takes 8/3 tosses on average).

BIASED COIN

If the coin was biased, TH and HT would occur with equal probability.

So you could assign THHT, HTTH and THTH to the three choices, with other 4-toss outcomes rejected.

Or you could assign HTT, THT and TTH to the three choices, with other 3-toss outcomes rejected.

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