wo identically looking black boxes hold two balls each. One of the boxes
holds two black balls and the other box holds one black and one white ball. We
randomly fix our attention on one of the boxes. Then we repeatedly take out
one of the balls at random and then put it back. Suppose we did it
times and each time this happens to be a black ball. What is the probability
to recover the white ball at the
-rd
draw?
We introduce a random variable
Denote as
the conditional distribution of
after the
-th
draw. The variable
(
=white
ball,
=black
ball) is the result of the
-th
draw.
Before the first draw we have the
distribution
Hence, by the formula (
Total probability
rule
),
We compute the
via the fromula
(
Inversion_remark
)
We repeat the procedure for the draw
2:
We finally compute the probability for the third
draw:
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