N percent of A’s are B’s. X is an A. This is all we know about the matter. Á It’s N percent probable that X is a B.
Prob of (A and B) = prob of A X prob of B.
Prob of (A XOR B) = prob of A + prob of B.
Prob of (A or B) = Prob of A + prob of B - prob of (A and B).
Prob of (A and B) = Prob of A X (prob of B after A occurs).
PLAYING. There are two possible outcomes: P1 (I win) and P2 (I lose). P1 is 1/36 likely and gains $3,536; P1 is worth (1/36 X $3,536) or $98.22. P2 is 35/36 likely and loses $100; P2 is worth (35/36 X -$100), or -$97.22. The expected gain of alternative P is ($98.22 - 97.22), or $1.
Most things true of X also are true of Y.
X is A. This is all we know about the matter. THUS: Probably Y is A.
A occurred more than once. B is the only additional factor that occurred if and only if A occurred. Á Probably B caused A, or A caused B.
A occurred in some case. B didn’t occur in the same case. THUS A doesn’t necessarily cause B.
A occurred in the first case but not the second. The cases are otherwise identical, except that B also occurred in the first case but not in the second. THUS Probably B is (or is part of) the cause of A, or A is (or is part of) the cause of B.
A changes in a certain way if and only if B also changes in a certain way. THUS Probably B’s changes caused A’s, or A’s caused B’s, or some C caused both.