Probability Basics

Section 6.2.1

Starter 6.2.1

• Roll two dice and record the sum shown.

• Repeat until you have done 20 rolls.

• Write a list of all the possible outcomes (sums) and how many of each you got.

• Report those totals to me when I call for them.– I will record class totals and show the

distribution of outcomes.

Objectives• Write the sample space for a set of events.

• Count the outcomes in a sample space by using the multiplication principle.

• Describe what is meant by the complement of an event.

• Describe what it means to say that two events are disjoint.– Use the addition rule for disjoint events to

answer “or” probability questions.

The Sample Space• The sample space of an event is a listing

of all possible outcomes.• List all the sums possible in the starter.

S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

• Suppose the question had asked you to do 20 rolls of two dice and note the individual dice face-up spots as ordered pairs– Example: (red 3, green 6)

• Write the sample space for rolling two dice– In other words, write an organized list or table

showing all the possible ordered pairs

The Sample Space of Two Dice1,1 1,2 1,3 1,4 1,5 1,6

2,1 2,2 2,3 2,4 2,5 2,6

3,1 3,2 3,3 3,4 3,5 3,6

4,1 4,2 4,3 4,4 4,5 4,6

5,1 5,2 5,3 5,4 5,5 5,6

6,1 6,2 6,3 6,4 6,5 6,6

• Notice that each of the 36 outcomes is equally likely• The sums are not equally likely because each sum has

different numbers of outcomes above• Compare theoretical probability of each sum with the

empirical probabilities we found in the starter

The Multiplication Principle

• If task A can be done in x ways, and task B can be done in y ways, then the number of ways to do both A and B is – There were 6 ways to roll the red die and 6

ways to roll the white die, so there are 36 ways to roll both dice.

– In yesterday’s starter, there were 2 ways to toss the first coin and 2 ways to toss the second coin and 2 ways to toss the third coin, so there are 8 ways to toss the three coins.

x y

The Rules of Probability• Rule 1: For any event A:• Rule 2: If S is the sample space: P(S) = 1• Rule 3: The complement of event A is the event

that A does not occur– The Complementary Probability Rule is:

P(Ac) = 1 – P(A)

• Rule 4: Two events A and B are disjoint if they are mutually exclusive (if A happens, B cannot and vice-versa). – The probability of two disjoint events is:

P(A or B) = P(A) + P(B)• This is called the “disjoint or” rule

0 ( ) 1P A

Disjoint Examples• Roll two dice. I will pay you $1 if you get a sum

of 7 or 8.– What is event A?– What is event B?– Are A and B disjoint?– What is P(A or B)?

• Choose a student from this class at random. I will pay you $1 if the student is a senior or a boy.– What is event A?– What is event B?– Are A and B disjoint?

Objectives• Write the sample space for a set of events.

• Count the outcomes in a sample space by using the multiplication principle.

• Describe what is meant by the complement of an event.

• Describe what it means to say that two events are disjoint.– Use the addition rule for disjoint events to

answer “or” probability questions.

Homework

• Read pages 318 – 329

• Do problems 9, 12, 15, 18, 20-23