Extremely Difficult Probability Problem

[KJL]

This is actually what I thought it was initially, but after re-reading the orgiinal post, that was not what the question asked. 1/1024 does not take into account the times when there is one white ball in the box that you keep picking.

[AaronBrown]

Here are the numbers. The first column is the number of white marbles, the second is the probability of this result given the coin-flipping procedure, the third is the probability of getting 10 out of 10 white marbles given the number of white marbles in the urn; and the last number is the product of the second and third.

Adding up the numbers in the fourth column gives 0.013913. This is the probability of drawing 10 out of 10 white marbles, given the conditions. 0.000977 in the bottom right hand cell of the table is the probability of getting 10 white marbles in the urn, dividing by 0.013913 gives 0.070190.

0 0.000977 0.000000 0.000000
1 0.009766 0.000000 0.000000
2 0.043945 0.000000 0.000000
3 0.117188 0.000006 0.000001
4 0.205078 0.000105 0.000022
5 0.246094 0.000977 0.000240
6 0.205078 0.006047 0.001240
7 0.117188 0.028248 0.003310
8 0.043945 0.107374 0.004719
9 0.009766 0.348678 0.003405
10 0.000977 1.000000 0.000977