What is the probability that it rained during the race

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

P(R) = .18

P(~R) = 1-.18 = .82

P(H/~R) .05

P(H/R) = .35

P(R/H)=?

Use Bayes' Theorem

P(R/H)P(H) = P(H/R)P(R)

= .35(.18)

= .063

P(R/H) = .063/P(H)

= .063/(P(H/R)P(R) + P(H/~R)P(~R))

= .063/(.35(.18) + .05(.82))

= .063/.104

= ..60576923

P(R/H) = about 61%

Answer 2

If you make a tree diagram, I think it helps:

Dwayne can get hurt if it rains (probability = 0.063) or if it does not rain (probability = 0.041). Adding these together, and we see that the probability that Dwayne gets hurt (ignoring the weather) is 0.104

The probability that it rained and he got hurt is 0.063/0.104 = 0.6058 = 61%

The probability that it did not rain and he got hurt is 0.041/0.104 = 0.3942 = 39%

So the probability it rained was 61%

You can also make a table:

Rain No Rain Total

Gets Hurt 6.3 4.1 10.4

Not Hurt 11.7 77.9 89.6

Total 18 82 100