Paul W. answered • 25d
Mastering Math and Stats can be EASY!! Let me show you how...
Good question! Let's see if I can give you a good answer:
Suppose for this problem that Pr[E] = [ 5/8] and Pr[F] = [ 3/4]. Just as in the book, Pr[(E ∪F)′] = 0.
(1) What is Pr[E | F]?
To get the conditional probability of E given F, we need to find the Pr[(E∪F)]. We know that the complementary probability Pr[(E ∪F)′] = 0 which implies that Pr[(E ∪F)] = 1. This does not make sense because it means that E and F perfectly overlap. There must be some piece of information that is missing from the book to get the correct answer.
If Pr[(E ∪F)] exists and is not 1 then we can come up with an answer for both pieces. Let's say that the original question had Pr[(E ∪F)′] = 0.8. Then we can use the general rules for probability to find
Pr(E | F) as Pr[(E ∪F)] / Pr(F) = 0.2/0.75 = 0.267
(2) What is Pr[F | E]?
The probability of F given E may be found the same way only now we are conditioning on E so Pr(E) will be in the denominator. The numerator doesn't change because intersection of events is commutative, which is to say that E and F is the same as F and E. This may be demonstrated by looking at a table for E and F. If we find where E and F intersect on some value looking first at the E row and then finding the corresponding F column, we will find that we end up at the same place if we start with the column with the F value we care about and then moving down the column for the corresponding row in E.
Pr(F | E) may then be written as Pr[(E ∪F)] / Pr(E) = 0.2/0.675 = 0.296
It should be noted that there is a limiting amount for Pr[(E ∪F)′] where values smaller than 0.325 (3/8) give a nonsensical answer to the problem. If we picture a Venn diagram showing the overlap of E and F and assume that F completely contains E then the upper bound of Pr[(E ∪F)] must be 5/8. There is no way E and F can have a greater overlap than the minimum of E and F so we know that the non-overlapping part cannot be 0 or anything smaller than 3/8.
