Calculating conditional probabilities

2-2.4 Calculating conditional probabilities.

To assign P( A | M ) a value we must consider M as the new universe. Within that new universe we see just one A outcome, with a probability of 0.095. But a probability of 0.095 (with respect to the old universe where the total probability is one) counts for more in this new M universe, which has a total probability of 0.105. It seems natural to choose P(A | M ) = 0.095 ÷ 0.105, i.e., 0.905. The idea in general is

This is the inclusion of one set in another, but we may also rewrite this equation as a rule for the intersection of multiple sets in general, i,e., the logical AND:

P(A_i and M) = P(M) × P(A_i | M).

(NB: We use the subscript on A to specify which field the agreement probability refers to.) The AND rule says that if M occurs 10.5% of the time and A_i occurs 90.5% of those times, then A_i and M occur simultaneously 9.5% of the time. Because AND is symmetrical we can also conclude that

P(A_i and M) = P(A_i) × P(M | A_i)

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