Question

Off-road motorbike racing can be dangerous, especially when it rains. It has been found that it rains during

18% of races. When it does not rain, there is a 5% chance that Dwayne (a rider) will get hurt. However, when it

rains, the chance that he gets hurt increases to 35%. Dwayne has just finished a race and is hurt. What is the

probability that it rained during the race?

Answer 1

Let $R$ denote the event "rain during race". Let $H$ denote the event " a rider will get hurt".

Given $P(R)=0.18, P(H|R)=0.35, P(H|R^C)=0.05.$

$$P(R|H)=\dfrac{P(R)P(H|R)}{P(R)P(H|R)+P(R^C)P(H|R^C)}$$

$$=\dfrac{0.18(0.35)}{0.18(0.35)+(1-0.18)(0.05)}\approx0.6058$$