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Three coins are flipped. If at least one of them comes up heads, then what is the probability that they all come up heads?

The answer to this is 1/7, as it is the #(ways to have all heads)/#(ways to have at least one head).

If the question had asked, probability of all heads given that the first flip is heads, then the answer would have been 1/4 (because 1/2 * 1/2).

Why are these two answers different? Why is it that the "at least one" is a weaker condition, when at least simply means that one of the three coins is heads, which is the same as choosing a specific head, like the first coin? I would greatly appreciate some intuition here. Thanks!

Why is the at least one case less likely? If anything, shouldn't it be more likely because it doesn't restrict a specific coin to come up heads?

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    $\begingroup$ Suppose you had a thousand tosses. Saying that "at least one $H$" appears is practically meaningless (since it would be astonishing to throw $1000$ Tails in a row). Specifically, that assumption only rules out one possible sequence out of $2^{1000}$. But saying "the first toss was $H$" rules out half the total number of sequences. $\endgroup$
    – lulu
    Jun 13, 2020 at 17:56
  • $\begingroup$ @lulu : You should post that as an answer. $\endgroup$ Jun 13, 2020 at 19:18

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General result: Let $A, B, C$ be events in a probability space, all with positive probability. If $A\subseteq B\subseteq C$, then:

$P(A \mid B) \ge P(A\mid C)$.

This follows because $A$ 'takes up more of ' $B$ than it does of $C$.

If you apply the above general result to these events in the space of flipping three coins:

$A$ is the event of all three coming up heads;

$B$ is the event of the first coin coming up heads; and

$C$ is the event of at least one coin coming up heads;

you get that $A$ given $B$ is more likely than $A$ given $C$, as desired.

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  • $\begingroup$ OHHHH THANK YOU $\endgroup$ Jun 13, 2020 at 21:03
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$$ \begin{array}{c} & HHH \\ & HHt \\ & HtH \\ & tHH \\ & Htt \\ & tHt \\ & ttH \end{array} $$ The seven outcomes in which at least one "head" appears are listed above. In just one instance, all three are "heads."

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