Possion distribution exercise

Question

Suppose box A contains countless black balls and box B contains one white ball. We took the black ball out of box A, put it into box B, and experimented with selecting one ball randomly from box B. At this time, the number of black balls taken out of box A and carried to box B follows Poisson distribution.

How could I calculate the probability that the ball from B is white.

and how could I calculate if the ball from box B was white the probability distribution for the number of black balls moved to box B.

Answer 1

Some things to think about:

• The probability of picking a white ball out of box $B$ given we have $X+1$ balls in the box is $\frac{1}{X+1}$ where $X$ is the number of black balls in box $B$. This is because there is an equally likely chance of picking any ball out of the box (we assume).
• We are given $X$ follows a Poisson Distribution. Therefore, the probability of picking a white ball will not be constant, it will depend on the value $X$.
• Think about the probability distribution of $X$. Do you know what the formula is for a Poisson Distribution? Are you given any parameters that can allow you to find exact probabilities for picking a white ball when there is $X$ black balls also in the box?

In general, by the Law of Total Probability, the probability for picking a white ball out of the box will be

$$\sum_{i=0}^\infty \left( \mathbb{P}(X=i)\cdot\frac1{i+1} \right) =\sum_{i=0}^\infty \left( \frac{e^{-\lambda}\lambda^i}{(i+1)!}\right) = \frac{1-e^{-\lambda}}{\lambda} $$