A coin is tossed 5 times. What is the probability that tail appears an odd number of times?

Explanation

Let `X` denote the number of tails when a coin is tossed `5` times.
` {X} `follows a binomial distribution with` {n}=5 ; p=frac{1}{2} ; q=1-p=frac{1}{2}` Then `P(X=r)=^{5} C_{r}(frac{1}{2})^{r}(frac{1}{2})^{n-r}=^{5} C_{r}(frac{1}{2})^{5}`

The required probability `=P(X= odd )`

`=P(X=1)+P(X=3)+P(X=5) `

`=^{5} C_{1}(frac{1}{2})^{5}+^{5} C_{3}(frac{1}{2})^{5}+^{5} C_{5}(frac{1}{2})^{5})`

`=(frac{1}{2})^{5}[5+10+1] `

`=frac{16}{32} `

`=frac{1}{2}`