A computer lab has two printers. Printer I handles 40% of all the jobs. its printing time is Exponential with the mean of 2 minutes. Printer II handles the remaining 60% of jobs. Its printing time is Uniform between 0 minutes and 5 minutes. A job is printed in less than 1 minute. What is the probability that it was printed by printer I?
My attempt to solve this looks like the following (sorry, first attempt at LaTeX):
Given:
$P(Printer 1) = .4$
$P(Printer 2) = .6$
For Printer 1, $1/\lambda$ = 2 minutes. Therefore $\lambda = 1/2$.
$P(time < 1 minute | Printer 1) = F(1) = 1 - e^{-1/2} = 0.3935$
With printer 2, the std. dev. given is 5 minutes. 1 minute is .2 standard deviations from the mean of 0.
$P(time < 1 minute | Printer 2) = \Phi(0.2) = 0.5793$
By Bayes rule and the law of total probability, $P(Printer 1 | time < 1 minute) =$
$P(Printer 1) * P(time < 1 minute | Printer 1)\over P(Printer 1) * P(time < 1 minute | Printer 1) + P(Printer 2) * P(time < 1 minute | Printer 2)$
$=$
$0.4 * 0.3935 \over (0.4 * 0.3935+ 0.6 * 0.5793)$
$=$
$0.3117$
Only, the answer given is .567. I can't tell where I've gone wrong. Any hints (or even tips on better use of LaTeX) would be appreciated.
The probability the time is less than $1$, given it is printer $2$, is $\frac{1}{5}$.