A fair coin is flipped repeatedly What is the probability th

A fair coin is flipped repeatedly. What is the probability that the fifth tail occurs before the 10th head?

[Hint: Phrase this as a problem \"the 5th tail occurs on or before the kth flip\" for an appropriate k.]

Solution

It is clear that when p = q = 1/2, the formula for k successes in n trials is:

P[k] = nCk / 2^n

Now the 5th tail has to occur on the 5th or 6th or .... 14th trial, the last of which are 5 tails, 9 heads.

Now in such a sequence, the 5th tail must occur on the last trial with a probability of 1/2, so

desired probability is:

= (4c4/2^4 + 5c4/2^5 +6c4/2^6 + .....13c4/2^13) *(1/2) for last trial

= 0.91