" 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 . . .
The answer is "Yes".If the occurrence of makes more likely,
does the occurrence of make more likely?
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.
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!