Joe Galaxy is astronaut for project Pluto. Mission success or failure depends only on behavior of three major systems. Joe decides the following assumptions are authentic and apply to the performance of an whole mission: (1) The mission is a failure only if two or more of the major systems fail. (2) System I, the Wii system, will fail with the probability 0.15. (3) If at least one other system fails, no matter how this comes about, System II, the I-pad system, will fail with the conditional probability 0.5. If no other system fails, the I-pad system will fail with probability 0.1. (4) System III, the Beer Cooler, fails with the probability 0.5 if the Wii system fails. Otherwise, the Beer Cooler cannot fail.
a. Determine the probability that the mission succeeds but that the Beer Cooler fails?
b. Determine the probability that all three systems fail?
c. Given that more than one system failed, estimate the conditional probability that:
i. The Wii did not fail.
ii. The Beer Cooler failed.
iii. Both the Wii and the I-pad failed.
d. About the time when Joe was due back on Earth, you overheard the radio broadcast regarding Joe in a very noisy room. You're not positive what the announcer say, but, based upon all available information, you decide that it is twice as likely that he reported "Mission the success" as that he reported "Mission a failure." What now is the conditional probability (to you) that the Wii failed?