Thread: A problem come from "Introduction to Probability Models"

" If the occurrence of B makes A more likely, does the occurrence of A make B more likely?

I guess yes. But I could not figure out the reason. Thanks~~

Hello, shiningstarpxx!

An interesting question . . .

If the occurrence of makes more likely,
does the occurrence of make more likely?

The answer is "Yes".
We need Bayes' Theorem: .


We are told that, if happens, is more likely to happen.
. . That is, the probability of , given , is greater than the probability of .

In symbols: .

This means: .

Hence, we have: .


Therefore, the probability of , given , is greater than the probability of
. . The occurence of makes more likely.

Originally Posted by shiningstarpxx

" If the occurrence of B makes A more likely, does the occurrence of A make B more likely?

I guess yes. But I could not figure out the reason. Thanks~~

What you're asking is the following:

Does Pr(A | B) > Pr(A) => Pr(B | A) > Pr(B) ?

The answer is yes. Consider:

.

But it's given that .

Therefore and the implication follows.


Edit: Too fast for me this time, Soroban. But I like to think our replies ...... complement each other lol!