I have a sequence of $15$ experiments and I want to know the probability $S_1$ of success of each experiment so that the overall probability to obtain $10$ successes in the $15$ experiments run equals $P_1$. To do so, I use binomial distribution and I am able to determine $S_1$ quite easily.
Now I want to add $7$ more different experiments for a total of $22$ heterogeneous experiments and have an overall probability of obtaining $11$ successes of $P_2$. How can I determine the probability of success $S_2$ of the $7$ new experiments in order to have the probability $P_2$ of obtaining $11$ successes over $22$ experiments, given that the first $15$ experiments have success probability of $S_1$?