# Computing probability of completing a task composed of independent events

This is a general question. I have a task that is composed of 3 independent events: A, B, and C. All are mutually exclusive and don't happen at the same time. So first A, then B, then C. I know the probability of completing each event with respect to time, I have the pdfs. How can I calculate the probability of completing the task at hand with respect to time?

Since you have the pdf of each, what you want to calculate is the sum of the three random times: $T = T_A+T_B+T_C$. Depending on the exact form of your pdfs, you can try one of three approaches:
3. Analytically calculate the distribution of the sum by either multiplying thecharacteristic functions of each pdf and then back-transforming to a denstity, or by directly performing two convolutions: $f_C*(f_A*f_B)$