Each of boxes in a line contains one red marble, and for , the box in the position also contains k white marbles. A child begins at the first box and successively draws a single marble at random from each box in order. He stops when he first draws a red marble. Let P(n) be the probability that he stops after drawing exactly n marbles. The possible values(s) of n for which
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The probability of drawing exactly n marbles can be calculated as the product of drawing n − 1 white marbles followed by a red marble:
P ( n ) = 2 1 × 3 2 × ⋯ × n n − 1 × n + 1 1 = n ( n + 1 ) 1
Solving P ( n ) < 2 0 1 0 1 gives us
n ( n + 1 ) > 2 0 1 0 ⇒ n ≥ 4 5