A coin is flipped ten times What is the probability that the

A coin is flipped ten times. What is the probability that the first three flips are all HEADS given that an equal number of HEADS and TAILS are flipped? How does this conditional probability compare with the simple probability that the first three flips are HEADS?

Solution

So there are equal number of tails and heads ie 5 heads and 5 tails.

We can choose 5 places for heads in :C(10,5)=252 ways

Number of configurations where first three places are for heads are:

C(7,2)=21 as first three are fixed for heads so we need to choose 2 more from remaining 7 places

So probability is:21/252=1/12

Simple probabiilyt that first three flips are heads is:1/2^3=1/8