# Group aggregates of conditional probabilities

I suspect I don't know the academic or SEO word for what I'm looking for. Put simply, I want to know a group's total expected success rate given each member's likelihood of attempting the "trial" and each member's likelihood of success. Each member can have 0 or 1 attempts, and all are independent.

In the most trivial meaningful example, let's say member A and member B are the only ones in the group. Their probabilities look like, say

Member Chance of Attempt Chance of Success
A 0.4 0.95
B 0.1 0.99

I can straightforwardly enumerate the possible outcomes

Scenario Likelihood Expected Group Success Rate
Neither tries 0.54 N/A
Only A tries 0.36 0.95
Only B tries 0.06 0.99
Both try 0.04 0.97

Unless I'm mistaken, this is a pretty straightforward calculation to find the overall expected group success rate: $$(0.36 \times 0.95 + 0.06 \times 0.99 + 0.04 \times 0.97)/(0.36 + 0.06 + 0.04) = 0.957$$ which is NOT $$(0.4 \times 0.95 + 0.1 \times 0.99)/(0.4 + 0.1) = 0.958$$

As you might imagine, my dataset is much larger, and I can't enumerate all of these possible attempt scenarios. Is there a way to linearly combine elements of the first table to arrive at the solution?

• Welcome to CV, Brandan. Could you explain why you separate chances of attempts and successes? Why is that any different from more simply stating the chance of eventual success for each member? E.g., A has a $0.95\times 0.4$ chance and B has a $0.1\times0.99$ chance. Then, why do you wish to enumerate scenarios at all? Isn't your question more simply about the expected number of successes? That can be found by adding the expectations of each member (according to linearity of expectation).
– whuber
Commented Oct 31, 2023 at 13:34
• Thank you! I am not interested in the expected number of successes. I'm specifically interested in the success rate. I assume the "no attempt" scenario is what breaks the link in the relationship between expected number of successes and expected number of trials? Commented Oct 31, 2023 at 14:04
• Could you define what you mean by "success rate"?
– whuber
Commented Oct 31, 2023 at 14:24
• The number of successes divided by the number of attempts. Commented Oct 31, 2023 at 16:26
• But you don't describe any such data: you only describe various probabilities.
– whuber
Commented Oct 31, 2023 at 18:14