Chapter 5 Probability Rules

5.1 Multiplication Rule

In this section we again use the blood type data in Table 5.1 below.

Table 5.1: Frequencies by ABO and Rh Groups

Group	\(+\)	\(-\)
A	360	60
B	76	14
AB	25	5
O	390	70

In the previous section we computed the conditional probability

\[ \mbox{P(A | }-) = \frac{60}{149} = 0.403 \]

based on the data in Table 1.
The numerator 60 came from the number of Rh \(-\) people who are also ABO group A, and the denominator 149 is the total number of Rh \(-\) people. If we divide the numerator and denominator by 1000, then the quotient is not changed so that we have

\[ \mbox{P(A | }-) = \frac{60/1000}{149/1000} = \frac{0.060}{0.149} = \frac{\mbox{P(A and }-)}{\mbox{P(}-)} \]

This illustrates a general property of probability: If X and Y represent possible outcomes from a random event, then the probability of X given Y is

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

Multiplying on both sides of the above equation by P(Y) gives

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

This is called the multiplication rule, and it allows us to move back and forth between joint probabilities and conditional probabilities.

Let’s try this out, using Table 5.2 (which we saw previously).

Table 5.2: Probabilities by ABO and Rh Groups

Group	\(+\)	\(-\)
A	0.360	0.060
B	0.076	0.014
AB	0.025	0.005
O	0.390	0.070

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5.2 Exercises

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