(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

You flip a coin n times. The probabilty of getting a head on any flip is p. What is the probability that the number of heads flipped is always greater than the number of tails flipped?

3. The attempt at a solution

For example,

if n=1, the only possibility is H

if n=2, the only possibility is HH

if n=3, the two possibilities are HHH or HHT

if n=4, the three possibilites are HHHH, HHHT, or HHTH

if n=5, the six possibilites are HHHHH, HHHHT, HHHTT, HHHTH, HHTHH, and HHTHT

Of course you could keep doing this until you notice a pattern, take a guess at the formula, and then try to prove that forumula by induction. (I tried that approach but didn't get anywhere). Conditioning on the first flip clearly won't simplify the problem, and conditioning on the (n-1) flip or the nth flip doesn't seem to simplify matters either. I'm stumped.

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# Flipping a coin (similar to the ballot problem)

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