Let's say that I have a population that I can split into two subgroups, containing $60 \% $ and $40 \% $ of the population respectively.

And I know the probability of some event in each of the two groups:

$P_1(A|B) = p_1$

$P_2(A|B) = p_2$

I want to calculate the probability of the event in the entire population. Can we simply assume that it will be:

$P(A|B) = 0.6p_1 + 0.4p_2$


You have probability of $A$ given $B$ and being member of a subpopulation $C$,

$$ P(A \mid B \cap C) = \frac{P(A \cap B \cap C)}{P(B \cap C)} $$

by the law of total probability

$$ P(A \mid B) = \sum_n P(A \mid B \cap C_n) \,P(C_n) $$

Notice that you incorrectly written it as $P(A \mid B)$, while it obviously depends also on group assignment $C$ if you have different probabilities per each group.


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