Thread: red marble

Each of 2010 boxes in a line contains a single red marble,
and for 1<(equal) k <(equal)2010,
the box in the kth position also contains k 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 P(n) be the probability that Isabella stops after drawing exactly n marbles. What is the smallest value of n for
which P(n) < 1/2010 ?

The answer is 45.



I first tried to figure out the probability for her to pick white marbles.

the first box 1/2
second box 2/3
third box 3/4
fourth box 4/5

I think I'm stuck....

Hint: Let's say the first red marble is drawn at draw number N. Then N > n exactly when the first n marbles drawn are all white. See if you can find a formula for P(N > n). Then see if you can use that formula to find a formula for P(N = n).