Question

Suppose that the longevity of a light bulb is exponential with a mean lifetime of eight years. If a bulb has already lasted 12 years, find the probability that it will last a total of over 19 years.

Short answer

The probability that it will last a total of over $19$years is$0.4169$.

Step-by-step solution

Given information 

The longevity of a light bulb is exponential with a mean lifetime of eight years.

Solution 

The required probability can be written as $P(x>19\mid x>12)$.

We need to use the memory less property of exponential distribution.

$P(x>t+r\mid x>t)=P(x>r)$

So,

$P(x>19\mid x>12)=P(x>12+7\mid x>12)$

$=P(x>7)$

$=1-P(x\le 7)$

$=1-\left(1-e^{\frac{-1}{8}\times 7}\right)$

$=e^{\frac{-1}{8}\times 7}$

$=0.4169$