Problem:
Each of boxes in a line contains a single red marble, and for , the box in the position also contains white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let be the probability that Isabella stops after drawing exactly marbles. What is the smallest value of for which ?
Answer Choices:
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B.
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E.
Solution:
If Isabella reaches the box, she will draw a white marble from it with probability . For , the probability that she will draw white marbles from each of the first boxes is
so the probability that she will draw her first red marble from the box is . The condition is equivalent to , from which and . The smallest positive odd integer whose square exceeds is , and the corresponding value of is .
The problems on this page are the property of the MAA's American Mathematics Competitions