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A bag contains five white balls and three black balls. Your goal is to draw two black balls.

You simultaneously draw two balls at random. Once you have drawn two balls, you put back any white balls, and redraw so that you again have two drawn balls. What is the probability that you now have two black balls? (Include the probability that you choose two black balls on the first draw.)

 Apr 19, 2023
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There are two cases:

Case 1: You draw two black balls on the first draw. The probability of this is 3/8*2/7 = 3/28.

Case 2: You draw one black ball and one white ball on the first draw. The probability of this is 3/8*5/7 = 15/56​. You then put the white ball back in the bag and draw two more balls. The probability of drawing two black balls in this case is 3/8*2/7​=3/28​.

The total probability of drawing two black balls is 3/28 + 15/56 = 45/56​​.

 Apr 19, 2023

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