Q:

Extract balls from urn probability

We have two urns, the first with 6 white balls and 7 black balls and the second with 10 white balls and 5 black balls.
We extract a ball from the first urn and introduce it into the second one, then we extract from the second urn 5 balls,reintroducing them back after each extraction.Whats the probability all the 5 balls are white? What scheme could be used here? Is it Poisson and if yes how to use it given the fact that theres and extraction with replacement?

A:

Case 1) We initially extracted a white ball with probability $\frac{6}{13}$. Then the second urn has $11$ white balls and $5$ black balls. The probability that all $5$ selected are white is then $$\left(\frac{11}{16}\right)^5$$
Case 2) We initially extracted a black ball with probability $\frac{7}{13}$. Then the second urn has $10$ white balls and $6$ black balls. The probability that all $5$ selected are white is then $$\left(\frac{10}{16}\right)^5$$
All together we get 
$$\left(\frac{6}{13}\cdot\left(\frac{11}{16}\right)^5\right)+\left(\frac{7}{13}\cdot\left(\frac{10}{16}\right)^5\right)\approx0.1222$$

