Chapter 6 Bayes Rule

6.1 Restricting the Population

Reversing the order of X and Y above gives

\[ \mbox{P(Y and X)} = \mbox{P(Y | X)}\mbox{P(A)} \]

In the blood type example, it is clear that P(Female and Type O) is the same as P(Type O and Female). This is true in general: P(A and B) = P(B and A), it follows that \[ \mbox{Pr(A | B)}\mbox{Pr(B)} = \mbox{Pr(B | A)}\mbox{Pr(A)} \] so that \[ \mbox{Pr(A | B)} = \mbox{Pr(B | A)}\frac{\mbox{Pr(A)}}{\mbox{Pr(B)}} \] This formula is called \(\textbf{Bayes Rule}\). Applied to our earlier population, we have \[ \mbox{Pr(Type O | Female)} = \mbox{Pr(Female | Type O)}\frac{\mbox{Pr(Type O)}}{\mbox{Pr(Female)}} \]

Two outcomes A and B are \(\textbf{independent}\) when \(\mbox{Pr(A | B)} = \mbox{Pr(A)}\), which implies that knowing if B has occurred has no effect on the probability of A.

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6.2 Exercises

1. \(\text{Put exercises here}\)