Conditional Probability Issue?

Question

Sample size=entire data set. The probability of A=55.6%, B=52.3%, and C=52.4%. The probability of AB=35.4%, and BC=39.7%. Finally, the probability of ABC=30.3%.

Assuming that the various events are "dependent", is the "conditional probability" of ABC, given G (30.3/55.6=54.5%)? Similarly, is the "conditional probability" of ABC, given AB (30.3%/35.4%=85.6%)?

Thank you in advance.

Obviously I mean "A" and not "G" in the question below. That is, is the conditional probability of ABC, give "A."

Answer 1

Dear Jesse,

Yes, your numbers appear correct (at least to the precision that you give them).

To see it, take the definition of conditional probability: P(X | Y) = P(X ^ Y) / P(Y).

So with X = ABC and Y = A, you have P(ABC | A) = P(ABC ^ A) / P(A).

You already know P(A) = 0.556, so you need to evaluate the numerator P(ABC ^ A). The event (ABC ^ A), or the intersection of ABC with A, is simply ABC (you can easily see this in Venn diagram). Thus the numerator is another thing you already know, P(ABC) = 0.303 .

Thus, P(ABC | A) = P(ABC ^ A) / P(A) = P(ABC) / P(A) = 0.303 / 0.556 = 0.545 (to three decimal places), or 54.5% .

Likewise, P(ABC | AB) = P(ABC ^ AB) / P(AB) = P(ABC) / P(AB) = 0.303 / 0.354 = 0.856 (to three decimal places), or 85.6% .

And you are welcome.