Question 1) Please solve the following questions using classical probability theory. Show your math or describe your answer clearly.

a. Given P(A) = 0.6 ; P(B) = 0.5; and P(A ⋂ B) = 0.2 , then P(A|B) ?

b. If P(A) = 0.3 and P(B|A) = 0.3, are events A and B independent? If not why?

c. In May in Belgium, the high temperature exceeded 700 F on 20 days. It rained on 8 days when the high temperature was over 700 F. If we choose any day in May, the probability that in rained on that day is independent of the probability was over 700 F. Therefore, how many days did it rain in May?

d. Hex Yahtzee is a game in which each person rolls six dice at once. The dice are ordinary 6- sided cubes, with a different number on each face. If exactly five of the six dice come up the same, the roll is called five-of-a-kind, and score very well. What is the probability of rolling five-of-a-kind on your first roll in Hex Yahtzee?

Question 2) You work at company and are searching for the Higgs boson. You have one trillion particle traces from the Company’s Large Hadron Collider. You believe that ten thousand of the particles are Higgs bosons. Each particle trace is fed into the expert software. If a particle is a Higgs boson, the software will identify it correctly with a 0.9 probability. If it is not a Higgs boson, the software will reject it with a 0.99 probability.

If a particle is identified by HiggsView as a Higgs boson, what is the probability that it actually is a Higgs boson?

Plus, Questions 3 to 5 attached