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I have a (possibly-) unfair coin, which lands heads with probability $p$ and lands tails with probability $1-p$. I toss it twice; the tosses are independent and identically distributed.

The probabilities of the various outcomes will vary depending on $p$. For example, if I know that $p = \frac12$, then the probability of tossing two heads, which I'll abbreviate $\mathbb P(HH)$, can be calculated as $p^2 = \frac14$. Symmetrically, the probability of tossing two tails $\mathbb P(TT) = \frac14$.

On the other hand, if I know that $p > \frac12$, e.g. $p = \frac1{\sqrt 3}$, then $HH$ becomes more likely: $\mathbb P(HH) = \frac13$. Correspondingly, $TT$ becomes less likely: $\mathbb P(TT) = (1 - \frac1{\sqrt 3})^2 \approx 0.18$.

It seems like I can push $HH$ up above $\frac14$, but only by pushing $TT$ down below that same number. With that in mind, the following question might seem surprising:

Under what circumstances does $\mathbb P(HH) = \mathbb P(TT) = \frac13$?

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    $\begingroup$ Is this a trick question? Your statement "the tosses are independent and identically distributed" makes it mathematically impossible to have both probabilities equal 1/3. $\endgroup$ – Gamow Dec 2 '15 at 15:19
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    $\begingroup$ @Gamow: I believe my statement is accurate. This wouldn't be an interesting puzzle if it weren't for the apparent contradiction :) $\endgroup$ – Ben Millwood Dec 2 '15 at 15:26
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    $\begingroup$ I'll think about how to give subtle hints if no-one has got closer to the answer by tomorrow. $\endgroup$ – Ben Millwood Dec 2 '15 at 16:00
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    $\begingroup$ Must the coin have two faces, one with a single head and the other with a single tail? $\endgroup$ – Lopsy Dec 2 '15 at 16:24
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    $\begingroup$ Yes. (Actually, until you mentioned it I didn't deeply think about how you'd even go about weighting a coin to change the outcome of its probabilities. Probably the actual easiest thing would be to put a bunch of heads and tailses on a die and weight that appropriately. But that's irrelevant to the puzzle: from the puzzle's perspective the only thing that matters is that there are exactly two possible outcomes of each individual coin toss.) $\endgroup$ – Ben Millwood Dec 2 '15 at 16:35
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$p$ is a random variable, chosen with uniform probability over the interval [0,1]. The probability of two heads is $\int_0^1 p^2=\frac{1}{3}$, and by symmetry so is the probability of two tails.

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    $\begingroup$ Now that I think about it this makes perfect sense. There was definitely a reason why Ben Millwood said the flips were identically distributed. $\endgroup$ – orp Dec 2 '15 at 19:39
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    $\begingroup$ Also explains why they are possibly unfair. $\endgroup$ – Fimpellizieri Dec 2 '15 at 19:46
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    $\begingroup$ Of course, this destroys independence, though. In this case, the flips are only conditionally independent, conditional on the value of $p$. $\endgroup$ – user2357112 Dec 2 '15 at 20:10
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    $\begingroup$ @orp: No, it's about whether one event tells you something about another event. Also, you can't just suddenly fix $p$ like that. Let me put it this way. Let $H?$ be the event that the first coin lands heads, and let $?H$ be the event the second coin lands heads. In the sample space of all possible values of $p$, flip 1, and flip 2, $p(H?) = 1/2$ and $P(HH) = 1/3$, so $P(?H|H?) = P(HH)/P(H?) = (1/3)/(1/2) = 2/3 \ne P(?H)$. Thus, by the definition of independence, $?H$ and $H?$ are not independent. $\endgroup$ – user2357112 Dec 2 '15 at 21:28
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    $\begingroup$ A generalization to n coins. $\endgroup$ – xnor Dec 3 '15 at 7:05
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This seems to be mathematically impossible. The probability of two heads is $1/3$ and the probability of two tails is $1/3$.

Since they're equal, this gives us $$ p^2 = (1-p)^2 $$ $$ p = 1-p $$ $$ p = 1/2 $$

But $(1/2)^2 = 1/4$, not $1/3$.

If there really is a solution to this I'd be really interested to see it.

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    $\begingroup$ I agree that it seems impossible, but I insist that it is not! $\endgroup$ – Ben Millwood Dec 2 '15 at 15:54
  • $\begingroup$ @Ben Millwood Just to clarify that this isn't a trick question, are you insisting that it is mathematically possible to answer the question, or that it is mathematically possible for P(HH) = P(TT) = 1/3? If this is the dirty trick I'm suspecting, "Under no circumstances" is a potentially valid answer. $\endgroup$ – Ninety-Three Dec 2 '15 at 18:08

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