# 4.3 - Multiplication Rule

4.3 - Multiplication Rule
Multiplication Rule

The probability that two events A and B both occur is given by:

$$P(A\cap B)=P(A|B)P(B)$$

or by:

$$P(A\cap B)=P(B|A)P(A)$$

## Example 4-4

A box contains 6 white balls and 4 red balls. We randomly (and without replacement) draw two balls from the box. What is the probability that the second ball selected is red?

We'll see calculations like the one just made over and over again when we study Bayes' Rule.

## The Multiplication Rule Extended

The multiplication rule can be extended to three or more events. In the case of three events, the rule looks like this:

$$P(A \cap B \cap C)=P[(A \cap B) \cap C)]=\underbrace{P(C | A \cap B)}_{a} \times \underbrace{P(A \cap B)}_{b}$$

$$\text { But since } P(A \cap B)=\underbrace{P(B | A) \times P(A)}_{b}\colon$$

$$P(A \cap B \cap C)=\underbrace{P(C | A \cap B)}_{a} \times \underbrace{P(B | A) \times P(A)}_{b}$$

## Example 4-5

Three cards are dealt successively at random and without replacement from a standard deck of 52 playing cards. What is the probability of receiving, in order, a king, a queen, and a jack?

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