Exponential Distribution - Probabilities I'm trying to figure out the answer to the following question on a past exam given for practice. Since there are no solutions, I was hoping I could get the help needed to figure it out.

A BS1300 big-screen TV may die due to screen failure or power supply failure.
The time until power supply failure is exponential with parameter 0.00001.
The time until screen failure is exponential with parameter 0.00002. The time is in hours.
a) Which failure is more probable to occur first?
For this problem, I simply got the expected values and whichever was lower, it should be more probable as the time expected for it to fail was lower.
$E(X) = 100000$ hours for power supply, $E(X) = 50000$ hours for screen failure. Therefore, screen is more likely to fail first.
b) What is the probability that a BS1300 will die within 10000 hours?
PS = power supply, SF = screen failure
$P(PS = 10000) = 0.000016374 P(SF = 10000) = 0.000025422$. If we add it up, we get PS or SF $= 0.000025422$.
c) If the BS1300 is known to be dead, what is the probability that its power supply failed?
Part c), I'm not sure how to figure this out. I tried drawing a venn diagram but I'm unsure how to do it for exponential. Help would be appreciated on how to solve this problem.

(a) correct
(b) wrong, first of all here you should use the CDF but not the PDF. Also you cannot simply add them up.
PG for power good, SG for screen good.
$P(PG)=1-p(PS)$, $P(SG)=1-P(SF)$. 
BS1300 is not dead, equivalent to both PG and SG. Since PG and SG are independent.
So $P(PG\cap SG)=P(PG)P(SG)=...$
