Question

The memers of a consulting firm rent cars from three rental agencies:
50% from agency X, 20% from agency Y and 20% from agency Z.
Form past experience, it is known that 9% of the cars from agency X need a service and tuning before renting, 12% of cars from agency Y need a service and tuning before renting and 10% of the cars from agency Z need a service and tuning before renting. If the rental car delivered to the firm need service and tuning, find the probability that agency Z is not to be blamed.

Answer 1

Let E: the car needs service and tuning. Let $E_1,E_2,E_3$ be the events that car is rented from agency X,Y,Z respectively.
So, $P\left(E_1\right)=50\%,P\left(E_2\right)=30\%,P\left(E_3\right)=20\%,P\left(E|E_1\right)=9\%,P\left(E|E_2\right)=12\%,P\left(E|E_3\right)=10\%.$
By Bayes' Theorem, $P\left(E_3|E\right)=\frac{P\left(E_3\right)P\left(E|E_3\right)}{P\left(E_1\right)P\left(E|E_1\right)+P\left(E|E_2\right)+P\left(E_3\right)P\left(E|E_3\right)}$
$\Rightarrow P\left(E_3|E\right)=\frac{\frac{20}{10}\times \frac{10}{100}}{\frac{50}{10}\times \frac{9}{100}+\frac{30}{10}\times \frac{12}{100}+\frac{20}{10}\times \frac{10}{100}}=\frac{20}{45+36+20}=\frac{20}{101}$
$\therefore P\left(\overrightarrow{E_3}|E\right)=1-P\left(E_3|E\right)=1-\frac{20}{101}=\frac{81}{101}.$