100 black or white marbles in a bag.?

Question

There are 100 black or white marbles in a bag. You pick out 100 marbles with replacement (so you pick out a marble then put it back in the bag then shake the bag then repeat until you have completed 100 repetitions. What is the probability that every marble in the bag is black given that each marble you pick out is black?

Dan · Accepted Answer

This isn't elegant, but does work.  You could have 1 through 100 black.

P(all black) = 1 - P(not all black) = 1 - P(99) - P(98) - ...P(1) = 1 - (.99)^100 - (.98)^100 ...

After a few terms, the values are so tiny that you can ignore them.

Answer (rounded) = 43%

EDIT: Assuming a 1/101 probability makes no sense and leads to a silly answer.

M3 · Answer

the # of black marbles could be 0 through 100, each assumed to have a probability of 1/101

if there were, say k black marbles, 
P[100 black] = (k/100)^100
and P[100 black | 100 black drawn] = 1/(1 + 0.01k^100)

so, summing up for k = 0 to 100, we get
indicated Pr = (1/101)*sum 1/(1+0.01k^100), k = 0 to 100
= 0.019704
---------------

Trending News

100 black or white marbles in a bag.?

2 Answers