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100AHW3S

# 100AHW3S - STAT 100A HWIII Solution Problem 1 Suppose an...

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STAT 100A HWIII Solution Problem 1: Suppose an urn has r red balls and b blue balls. We random pick a ball, and then we put three balls of the same color back to the urn. After that we randomly pick a ball again. (1) What is the probability that the second pick is red? A: P (2nd = red) = P (1st = red) P (2nd = red | 1st = red) + P (1st = blue) P (2nd = red | 1st = blue) = r r + b r + 2 r + b + 2 + b r + b r r + b + 2 = r r + b . (2) Optional: Suppose after the second pick, we again put three balls of the same color back to the urn, and then we randomly pick a ball again. What is the probability that the third pick is red? (The optional questions will be graded but will not be counted to the total score of your homework. They are just for fun.) A: P (3nd = red) = P (1st = red) P (2nd = red | 1st = red) P (3rd = red | 2nd = red 1st = red) + P (1st = red) P (2nd = blue | 1st = red) P (3rd = red | 2nd = blue 1st = red) + P (1st = blue) P (2nd = red | 1st = blue) P (3rd = red | 2nd = red 1st = blue) + P (1st = blue) P (2nd = blue | 1st = blue) P (3rd = red | 2nd = blue 1st = blue) = r r + b r + 2 r + b + 2 r + 4 r + b + 4 + r r + b b r + b + 2 r + 2 r + b + 4 + b r + b r r + b + 2 r + 2 r + b + 4 + b r + b b + 2 r + b + 2 r r + b + 4 = r

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