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The probability that a tossed coin lands heads is $p$. When the coin is tossed twice, it is equally likely that both tosses are the same and that the tosses are not both tails. What are the possible values of $p$, and which of these values is plausible for a physical coin?

From what I understand, getting $Pr(Head)=p$, so then $Pr(Tail)=1-p$.

This part I cannot understand:

When the coin is tossed twice, it is equally likely that both tosses are the same and that the tosses are not both tails.

Does this mean that both tosses are tails?

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The stated equality means $$P(HH+HT+TH)=P(HH+TT)$$ $$P(HT+TH)=P(TT)$$ $$2p(1-p)=(1-p)^2$$ Either $p=1$ or $$2p=1-p\implies p=\frac13$$ So $p=1\lor p=\frac13$, but only $p=\frac13$ is plausible for a physical coin.

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  • $\begingroup$ I see, thanks a lot for clarifying the question. $\endgroup$ – phoxd Sep 13 '17 at 4:06

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