Joe is about to take one of three tests: Quantum Field Theory (QFT), General Rel

Question

Joe is about to take one of three tests: Quantum Field Theory (QFT), General Relativity (GR), or Writing in Terminology that’s Fun (WTF). Joe has a 20% chance of sitting next to his (much smarter) friend Jan. If he sits next to Jan, he has a 45% chance of passing QFT which he will have a 25% chance of taking; a 55% chance of passing GR, which he will have a 50% chance of taking; and an 80% chance of passing WTF, which he will have a 25% chance of taking. If he does not sit next to Jan, he has a 30% chance of passing QFT, which he will have a 20% chance of taking; a 40% chance of passing GR, which he will have a 50% chance of taking; and a 75% chance of passing WTF, which he will have a 30% chance of taking. What is the probability that Joe passes the test?

Explanation / Answer

Case 1: Joe sits next to Jan p = 0.20

Then his probability of passing QFT = 45%, GR = 55% and WTF = 80%

and chances of taking the exams are 25%, 50% and 25% respectively

So, probability of passing the test =0.25*0.45 + 0.5*0.55 + 0.25*0.8 = 0.5875

Case 2: Joe doesn't sit next to Jan p = 0.8

As above, probability of passing the test = 0.3*0.2+0.4*0.5+0.75*0.3 = 0.485

Hence probability of passing the test = P(Case 1)*P(Passing|Case 1) + P(Case 2) * P(Passing| Case 2) = 0.2*0.5875 + 0.8*0.485 = 0.5055

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