# Ch 3.3 Addition and Multiplication Rule

Addition Rule are used to find “OR” in a procedure.

P(A or B) = P(A) + P(B) – P(A and B)

If A and B are mutually exclusive: P(A and B) = 0

P(A or B) = P(A)+ P (B) when A, B are mutually exclusive.

Ex1.  Toss a 6-face die once, use addition rule method to find  P(one or odd).

P(one or odd) = P(one) + P(odd) – P(one and odd)

= 1/6 + 3/6 – 1/6 = 3/6  =0.5

Ex2.  Toss a 6-face die once, use addition rule method to find P( one or even)

Because one and even or mutully exclusive, so P (one or even) = P(one) + P(even)  =  1/6 + 3/6 = 0.667

Ex3. Use the contingency table below:

GT = 51

Use addition rule to find P(male or iPhone).

P(male or iPhone) = P(male) + P(iPhone) – P(male and iPhone)   = 21/51 + 42/51 – 18/51 = (21+41-18)/51 =45/51 = 0.8823

### Multiplication Rule:

Multiplication Rule is used to find probability of two events: A and B.

$$\text{P(A and B)} = P(A) * P(B|A)$$

If A and B are independent, P(B|A) = P(B) so

$$\text{P(A and B)} = P(A) * P(B)$$ when A, B are independent.

A result of the multiplication rule gives the formula for conditional probability as:

$$\\text{P(A given B)}=\text{P(A | B)} =\frac{\text{A and B}}{P(B)}$$

Ex1: Given the two-way table below:

Find P( male |iPhone) =  P( male and iPhone)/P(iPhone) =  $$\frac{18/51}{42/51} = \frac{18}{42} = 0.4286$$