# Three people each flip two fair coins.Find the probability that exactly two of the people flipped one head and one tail.

Three people each flip two fair coins.Find the probability that exactly two of the people flipped one head and one tail.

Out of three persons,two persons can be chosen in $\binom{3}{2}$ ways.Each person flips two fair coins.So each persons gets $HH,HT,TH,TT$.Probability of a person getting one head and one tail is $\frac{1}{2}$.

So the probability that exactly two of the people flipped one head and one tail is $\binom{3}{2}\times\frac{1}{2}\times\frac{1}{2}=\frac{3}{4}$

But my answer is wrong.What is wrong in my approach.Please help me.

• Multiply by $1/2$ for the third person to flip double head or double tail. – André Nicolas Dec 1 '15 at 3:45

## 2 Answers

As you say, the probability of a person getting one head and one tail is $\frac12$. The probability of getting two heads or two tails is $\frac12$. Altogether, we'll have:

$${3 \choose 2}\times\left(\frac12\right)^2\times\frac12 = \frac38.$$

The $\left(\frac12\right)^2$ accounts for the two people who get a head and a tail, and the final $\frac12$ accounts for the one person who doesn't.

You accounted for the first person flipping a head and a tail, the second person flipping a head and a tail, but you didn't account for the third person not flipping a head and a tail.