Chapter 12.4, Problem 33E

To determine

To prove: That the probability that the chip will fail within the next $b$ years is

$\frac{\mathrm{Pr}(a\le X\le a+b)}{\mathrm{Pr}(a\le X)}$, where a variable $X$ denotes the time to failure (in years) of a computer chip which has been operating properly for $a$ years, and this probability equals $\mathrm{Pr}(0\le X\le b)$, and in this case, $X$ is an exponential random variable with the density function $f(x)=k{e}^{-kx}$.

Exponential random variables are said to be memoryless.