Question

A coin weighted so that $P\left(H\right)=2/3$ and $P\left(T\right)=1/3$ is tossed. If head appears, then a number is selected at random from the number $1$ through $9$, If tail appears, then the number is selected at random from the numbers $1$ through $5$. Find the probability $p$ that an even number is selected?

Answer 1

Probability of getting an even no, in $1$ to $9=\frac{4}{9}$

Probability of getting even no, in $1$ to $5=\frac{2}{5}$

True diagram is shown:

Hence,$P\left(E\right)=\frac{2}{3}.\frac{4}{4}+\frac{1}{3}.\frac{2}{5}=\frac{8}{27}+\frac{2}{15}=\frac{120+54}{405}=\frac{174}{405}$