2010 AMC 12A Problems/Problem 19
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 ?
Solution
The probability of drawing a white marble from box is , and the probability of drawing a red marble from box is .
To stop after drawing marbles, we must draw a white marble from boxes and draw a red marble from box Thus,
So, we must have or
Since increases as increases, we can simply test values of ; after some trial and error, we get that the minimum value of is , since but
See also
2010 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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All AMC 12 Problems and Solutions |
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