A jar contains one hundred marbles, each of which may be white or black. You pull out 100 marbles with replacement, and they are all white. What is the probability that all one hundred marbles are white?
(Note: “With replacement” means you take out a random marble, look at its colour, then put that marble back. Then repeat.)
You cannot calculate the probability because you don’t know how many of the marbles are white and how many are black.
There is at least one white marble, but they could all be white.
When you know the amount of white marbles (w) then the probability is (w/100)^100.
Let B = No of black marbles in the jar
If B = 1, Pr(observations) = (99/100)^100 = 0.366
If B = 2, Pr(observations) = (98/100)^100 = 0.133. etc. etc.
Summing all these probabilities from B = 1 to B = 99 gives 0.572 which is probability of observing no black marbles in 100 draws given any number of black marbles actually present in the jar.
Hence probability that jar has only white marbles is 1 – 0.572 = 0.428
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