Multiplication and Addition Rules for Probability

Special Topics

General Addition RuleLast time, we learned the Addition Rule for

Mutually Exclusive events (Disjoint Events). This was:

P(A or B) = P(A) + P(B). This is used when events A and B can’t happen at the same time, such as getting a head or a tail on a single coin flip. You can’t get both a head and a tail on the same coin flip!

There are other events which can happen at the same time, such as being a senior and being a student in Special Topics.

General Addition RuleLet’s let event A be the event that a person is

a senior. Let’s call P(A) = .25Let’s let event B be the event that a person is

in Special Topics. Let’s call P(B) = .8Let’s also call P(A and B) = .10If we go by the Addition Rule for Disjoint

Events, then P(A 0r B) = .25+.8 = 1.05 (what is wrong with this?).

The problem here is that a person can be both a senior and a Special Topics student at the same time. When this happens, things get counted twice!

A Diagram to Sort This Out!This is the situation in a Venn Diagram:

Notice the intersection gets counted twice! We need a formula that corrects for this.

General Addition RuleHere is the formula for the General Addition

Rule:P(A or B) = P(A) + P(B) – P(A and B). We

subtract one version of the joint probability since it gets counted twice.

When event are disjoint, the P(A and B) part of the formula equals 0. That takes us back to the formula for disjoint events: P(A or B) = P(A) + P(B).

ExampleMusical styles other than rock and pop

are becoming more popular. A survey of college students finds that the probability they like country music is .40. The probability that they liked jazz is .30 and that they liked both is .10. What is the probability that they like country or jazz? (Hint: a joint probability is given, so the General Addition Rule Applies!)

P(C or J) = .4 + .3 -.1 = .6

The Multiplication RuleWhat do we do to calculate the probability of

two (or more) independent events happening at the same time?

We use the Multiplication Rule for Independent Events. The key word is the word “and”.

The formula is: P(A and B) = P(A)P(B)What is the probability of flipping a coin and

getting a head, and rolling a die and getting a six?

1 1 12 6 12· =

IndependenceDefinition: Two events are independent when

knowing that one occurred does not change the probability that the other occurred.

Coin flips and dice rolls are independent.

HomeworkSection 8.1 Part 2 worksheet.