Question

The probability that an archer hits the target when it is windy is $0.4;$ when it is not windy, her probability of hitting the target is $\mathrm{0.7.}$ On any shot, the probability of a gust of wind is $0.3$. If the probability that she hits the target on first shot is $x,$ then the value of $100x$ is

Answer 1

Let $T$ be the event that archer hits the target
and $W$ be the event that it is windy.
Given : $P\left(T|W\right)=0.4,P\left(T|\overline{W}\right)=0.7$
$P\left(T\right)=P(T\cap W)+P(T\cap \overline{W})$
$=P\left(W\right)\cdot P\left(T|W\right)+P\left(\overline{W}\right)\cdot P\left(T|\overline{W}\right)$
$=\left(0.3\right)\left(0.4\right)+\left(0.7\right)\left(0.7\right)$
$=0.12+0.49=0.61$
$\Rightarrow 100x=0.61\times 100=61$