Question

5. The probability that an archer hits the target when it is windy is 0.4, and when it is not windy the probability of a hit is 0.7. On any shot, the probability of a gust of wind is 0.3. Find the probability that (a) the target is hit with a shot. (b) there was no gust of wind, assuming the target was missed.

Answer 1

Lets A denotes the event that the target is hit and B denotes the event that there is gust of wind.

Thus A' denotes the event that the target is missed and B denotes the event that there is no gust of wind.

Therefore, as given,

$P\left ( A|B \right )=0.4$

$P\left ( A|B '\right )=0.7$

$P\left ( B \right )=0.3$

(a) The required probability will be obtained as below:

$P\left ( A\cap B \right )=P(B)P\left ( A| B \right )=(0.3)(0.4)=0.12$

(b) The required probability will be obtained as below:$P\left ( B'| A' \right )=\frac{P\left ( A'\cap B '\right )}{P(A')}=\frac{P\left ( A'\cap B '\right )}{P\left ( A'\cap B \right )+P\left ( A'\cap B '\right )}=\frac{P(B')P\left ( A'| B' \right )}{P(B)P\left ( A'| B \right )+P(B')P\left ( A'| B' \right )}=\frac{(1-0.3)(1-0.7)}{(0.3)(1-0.4)+(1-0.3)(1-0.7)}=\frac{0.21}{0.18+0.21}=\frac{7}{13}$