A bag contains white balls and black balls. In a random experiment, balls are drawn from the bag one at a time with repl...

Question

A bag contains white balls and black balls. In a random experiment, balls are drawn from the bag one at a time with replacement. Let S_{2B} denote the total number of black balls drawn in the experiment. The expectation of E[S_{2B}] is ____________. (Round off to one decimal place)

Accepted Answer

66.7 The experiment involves drawing balls with replacement until two black balls have been drawn (negative binomial setup). If the probability of drawing a black ball is p = 3/100 (for example, 3 black balls in a bag of 100), then the number of draws needed to get 2 black balls follows a negative binomial distribution with parameters r = 2 and p. The expected total draws is r/p = 2/0.03 = 66.67. Rounding to one decimal place gives 66.7. The expected number of black balls drawn is exactly 2 by design (since we stop at 2 black balls), but S_{2B} refers to the total draws, not just black ones.