# Probability mass function of time of first head

This is an exercise from the probability book by Ross. This is not homework. Using conditional probability and the distribution of sum of two geometric random variables, the probability comes out to be $$\frac{1}{(n-1)}$$.

But I am not able to understand how can the same probability be deduced from just the hint without all the computations.

I am not sure but it seems to have to do with each of the previous $$(n-1)$$ tosses being equally probable of being the time of the first head. Is that correct?

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• One head occurs among the first $n-1$ flips. On which flip did it occur? Does the coin favor showing heads at any particular time compared to any other time?
– whuber
2 days ago
• Please add the self-study tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. yesterday
• @whuber As the coin is unbiased, all flips have the same probability of landing in heads. So no, the coin doesn't favor showing heads at any particular flip. yesterday
• @whuber I am confused regarding the distinction between the event of getting a head in the i-th flip and the i-th flip being the time of first head. yesterday
• Suppose the coin flips go T, H, T, H, .... You obtain a head on the i = 4th flip, but the first head was obtained on the i = 2nd flip.
– whuber
yesterday