Question

Jim remembers to re-torque his wheels 95% of the time. When he does remember there is only a 1% chance he will lose a wheel. When he doesn't remember there is a 15% chance he will lose a wheel. Jim lost a wheel. Determine the probability that he forgot to re-torque his wheels.

Answer 1

Let $R$ denotes the event "Jim remembers to re-torque his wheels".

Let $L$ denotes the event "Jim will lose a wheel".

Given $P(R)=0.95, P(L|R)=0.01,P(L|R')=0.15.$

$$P(R')=1-P(R)=1-0.95=0.05$$

By the Bayes' theorem

$$P(R'|L)=\dfrac{P(R')P(L|R')}{P(R)P(L|R)+P(R')P(L|R')}$$

$$P(R'|L)=\dfrac{0.05(0.15)}{0.95(0.01)+0.05(0.15)}\approx0.4412$$

The probability that Jim forgot to re-torque his wheels given that he lost a wheel is $0.4412.$