Problems in Probability

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Fall 2011 Strategic Practice 2: Section 3 (Thinking Conditionally) - Question 1
A bag contains one marble which is either green or blue, with equal probabilities. A green marble is put in the bag (so there are 2 marbles now), and then a random marble is taken out. The marble taken out is green. What is the probability that the remaining marble is also green?
Historical note: this problem was first posed by Lewis Carroll in 1893.
Solution: Let A be the event that the initial marble is green, B be the event that the removed marble is green, and C be the event that the remaining marble is green. We need to find P(C|B). There are several ways to find this; one natural way is to condition on whether the initial marble is green: P(C|B) = P(C|B,A)P(A|B) + P(C|B,Ac)P(Ac|B) = 1P(A|B) + 0P(Ac|B). To find P(A|B), use Bayes’ Rule: P(A|B) = P(C|B) = 2/3.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
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