# Calculate number of failures until success where the probability changes after a failure.

How do you calculate the expected value of the number of failures until the first success is reached where the probability will change after a failure. Let $$p$$ be the probability of success. Let $$X$$ be a random variable denoting the number of trials until success. Therefore, $$E[X] = \sum_{k=0}^{\infty} (1-\frac{m-1-k}{m})^{k}(\frac{m-1-k}{m})$$. Notice that subtracting $$k$$ will modify the probability after each failure.

• You need to state clearly exactly how you're supposing $\ p\$ to change after each failure. From the formula you give I'm guessing that $\ p\$ starts off at $\ \frac{m-1}{m}\$ and decreases by $\ \frac{1}{m}\$ after each failure. If that's correct, then $\ p\$ becomes negative after $\ m\$ successive failures, which is meaningless for a probability, and the probability of such a sequence of failures is $\ \frac{\left(m-1\right)!}{m^{m-1}}>0\$. – lonza leggiera Apr 23 at 7:37
• Also, the general term $\ \left(\frac{k+1}{m}\right)^k\left(1-\frac{k+1}{m}\right)\$ of your series diverges to $\ -\infty$ as $\ k\rightarrow\infty\$, and hence so does the series itself. – lonza leggiera Apr 23 at 7:37