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Suppose n identical fair coins are tossed. Those that show Heads are tossed again, and the number of Heads obtained on the second set of tosses defines a discrete random variable X. Assuming that all tosses are independent, find the probability mass function of X.

Answer: $$ f_X(x) = \binom nx \left(\frac 14\right)^x\left(1-\frac 14\right)^{n-x}. $$ (original imege: http://i.stack.imgur.com/IDUIJ.png)

I understand that the probability of flipping two heads is $1/4$ and the range of $X$ is $0$ up to $n$, so does this just come straight from the definition of a binomial distribution? Because the number of trials involved is not likely to be equal to $n$.

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up vote 5 down vote accepted

Hint: Modify the game slightly, so that whatever you get on the first toss of a coin, you toss the coin again. Then the number of heads in the original game is precisely the same as the number of coins that gave two heads in a row in the modified game. And the probability that a coin gives two heads in a row in the modified game is $1/4$.

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Look at the tosses for each coin. For coin $i$, consider "success" to be the event that the coin was tossed twice and that the second toss yielded heads. If the first toss was tails, then the outcome for that coin is already failure. You could equivalently think of this has the probability of tossing two heads in a row for coin $i$, where each coin is always tossed twice, but then counting how many coins showed heads both times.

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