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. 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
of recovering 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 formula
(
Inversion_remark
)
We repeat the procedure for the draw
2:
We finally compute the probability for the third
draw:
