Question

Suppose that of all used cars of a particular year 30% have bad brakes. You are considering buying a used car of that year. You take the car to a mechanic to have the brakes checked. The chance that the mechanic will give you the wrong report is 20%. Assuming that the car you take to the mechanic is selected at random from the population of cars of that year. The chance that the car's brakes are good, given that the mechanic says its brakes are good, is

• A. $\frac{28}{130}$
• B. $\frac{29}{31}$
• C. $\frac{37}{62}$
• D. $\frac{29}{62}$

Answer 1

The correct option is A $\frac{28}{130}$

Given $30$% of the cars of bad brakes

$P\left(E_1\right)=70$%$=\frac{7}{10}$ $P\left(E_2\right)=30$%$=\frac{3}{10}$

$\Rightarrow P\left(\frac{E}{E_1}\right)=0.2\times 0.2=0.04$

$\Rightarrow P\left(\frac{E}{E_2}\right)=0.8\times 0.8=0.64$

$\therefore P\left(\frac{E}{E_1}\right)=\frac{\frac{7}{10}\times \frac{2}{10}\times \frac{2}{10}}{\frac{7\times 4}{1000}+\frac{3}{10}+\frac{8}{10}+\frac{8}{10}}=\frac{28}{102+28}=\frac{28}{130}$

Hence, the answer is $\frac{28}{130}.$