Question

Suppose that two factories supply light bulbs to the market. Factory X's bulbs work for over $5000$ hours in $99\%$ of cases, whereas factory Y's bulbs work for over $5000$ hours in $95\%$ of cases. It is known that factory X supplies $60\%$ of the total bulbs available. What is the chance that a purchased bulb will work for longer than $5000$ hours?

• A. $\frac{876}{1000}$
• B. $\frac{544}{1000}$
• C. $\frac{974}{1000}$
• D. None of these

Answer 1

Answer: (C)

$\frac{974}{1000}$

Let $X$ be the event "comes from factory $X$" and $Y$ be the event "comes fom factory $Y$ " and Let $H$ be the event "works over $5000$ hours."

Therefore $P\left(X\right)=60\%=0.60\Rightarrow P\left(Y\right)=1-0.60=0.40$

Given that, Factory $X^{\prime }s$ bulbs work for over $5000$ hours in $99\%$ of cases.

$\therefore P\left(H|X\right)=99\%=0.99$

Also given, factory Y's bulbs work for over $$000$5000$ hours in %$95\%$ of cases.

$\therefore P\left(H|Y\right)=95\%=0.95$

Then by the Law of Total Probability we have

$P\left(H\right)=P\left(H|X\right)P\left(X\right)+P\left(H|Y\right)P\left(Y\right)$

$=\left(.99\right)\left(.6\right)+\left(.95\right)\left(.4\right)$

$=.974$

Thus $P\left(H\right)=0.974=\frac{974}{1000}$.