A Blarg Machine looks like a large gray metal box. On the front of the box is a

Question

A Blarg Machine looks like a large gray metal box. On the front of the box is a crank, and next to the crank is a small hole. Inside the box are n ping-pong balls, each with a distinct number from 1 to n. When you turn the crank, the Machine chooses one of the balls at random, and the chosen ball comes out of the hole. Now suppose you find a Blarg Machine. It contains either 100 balls, or 100,000 balls. Those are the only two possibilities, and they have equal probability. You turn the crank, and a ball comes out of the hole. The ball’s number is less than or equal to 100.

1. (10 points.) What is the probability that the Blarg Machine contains 100 balls?

2. (10 points.) What is the probability that the Blarg Machine contains 100,000 balls?

For full credit, you must answer both these questions using Bayes’ Rule. Also, note that if the ball’s number was greater than 100, then you would know with certainty that the Blarg Machine contains 100,000 balls—but that’s not what happened!

Explanation / Answer

Baye's theorem for conditional probability:

P(A|B) = P(A,B)/P(B) = P(B|A)*P(A)/P(B)

USING this:

1. P(Machine=100 | ball=within 100) = P(ball=within 100| Machine=100)*P(Machine=100) / P( ball=within 100)

P( ball=within 100) = P(ball=within 100| Machine=100)*P(Machine=100) +P(ball=within 100 | Machine=1000)*P(Machine=1000) = 1*.5+0.1*.5 = 0.55

THUS,

P(Machine=100 | ball=within 100) = 0.5/0.55=0.9091

2. P(Machine=1000 | ball=within 100) = P(ball=within 100| Machine=1000)*P(Machine=1000) / P( ball=within 100)

P( ball=within 100) = P(ball=within 100| Machine=100)*P(Machine=100) +P(ball=within 100 | Machine=1000)*P(Machine=1000)  = 1*.5+0.1*.5 = 0.55

THUS,

P(Machine=100 | ball=within 100) = 0.05/0.55=0.0909

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