I have the following problem:
Computers with a certain OS have a crash rate per month that is distributed as Poisson with $\lambda = 5$. After a patch is installed, the crash rate is distributed as Poisson with $\lambda = 3$ for 75% of computers, or is unchanged for 25% of computers.
A user installs the patch and reports 2 crashes the following month. How likely is it that the patch has taken effect? (His computer belongs to the 75%)
I've approached this problem as follows:
$P(Effect|Crashes = 2) = \frac{P(Crashes=2|Effect)*P(Effect)}{P(Crashes=2)} = \frac{P(Crashes=2|Effect)*P(Effect)}{P(Crashes=2 |Effect)*P(Effect)+P(Crashes=2 |No effect)*P(No effect)}$
Where
$P(Crashes =2 | Effect) = \frac{3^2 * e^{-3}}{2} \\ P(Crashes =2 | No effect) = \frac{5^2 * e^{-5}}{2}$
and the rest is substitution of values from the problem.
Is this correct?
Thank you