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Choosing Marbles

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #8808 Fun: (2.37) Difficulty: (2.45) Category: Probability Submitted By: lesternoronha1 Corrected By: MarcM1098

You choose one of two identical looking bags at random. One bag has three black marbles and one white marble. The other has three white marbles and one black marble. After choosing a bag you draw one marble out at random. You notice it is black. You then put it back and draw another marble out of the same bag at random. What is the probability that the second marble drawn is black?

The probability is 5/8.

The probability of event A happening given that event B already happened is the probability of A and B happening divided by the probability that B happened. This can be expressed as Pr(A|B)=Pr(A and B)/Pr(B).
In this case A is drawing a black marble and B is having already drawn a black marble.

Pr(A and B) = (1/2) * [(3/4)2 + (1/4)2] = 5/16.

Pr(B) = 1/2.

Pr(A|B) = Pr(A and B)/Pr(B) = (5/16)/(1/2) = 10/16 = 5/8.

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