Probability

Section 7.1

What is probability?

Probability discusses the likelihood or chance of something happening. For instance,

-- the probability of it raining tomorrow

-- the probability of rolling a 4 on a six-sided die

-- the probability of being struck by lightning

-- the probability of pulling an Ace out of a deck of cards

-- the probability of meeting your spouse on a blind date

We will try to answer some of these questions. Let’s start with a relatively easy example.

Consider rolling a six-sided die. What is the probability of getting a 4 on the die? Notice the notation we will use.

6

1)4( P

We can think of this as the ratio of successes to total

possibilities. There is 1 success (the 4) on a die and 6 possible sides.

“P of 4”

First, this does assume all sides are equally likely to come up when I roll the die. This is usually the case when we talk about dice.

This formula for probability will come in handy. Keep it in mind.

We should cover some terminology that we will use throughout our work here. Examples are given after each term.

Terminology

experiment: the act you do (roll a die)

outcome: any one possible result (1, 2, 3, 4, 5, or 6)

sample space: the set of all possible outcomes

( {1, 2, 3, 4, 5, 6} )

event: a particular set of outcomes ( {2, 4, 6} or evens)

trial: one instance of the experiment (If you roll a die 50 times, you’ve done 50 trials.)

success: specific event you’re interested in (If we want the probability that the die will be even, we have three successes; they are 2, 4, and 6.)

Theoretical probability: the probability of an event based on the context of the problem

Experimental probability: the probability of an event based on doing the experiment many times

Consider rolling a die. The theoretical probability of

getting an even number is . This is

gotten by thinking about the die and figuring that there are 3 successes and 6 total possibilities.

If we were to roll a die 100 times and actually roll an even number 47 of the times, our experimental probability

would be .

63)( evenP

10047)( evenP

Worksheets

“Probability applet activity” investigates the relationship between theoretical and experimental probability via an online activity. It simulates the tossing of a coin many times. You are looking at the proportion of heads.

“The difference between OR and AND” covers some fundamentals needed to understand probability.

“Probability experiment” is a group activity where you will think up a probability question and find its experimental probability.

Other optional worksheets are available online.

Consider the experiment

Roll die, toss coin.

What’s P(4 and H)?

Sample space: 1T 1H

2T 2H

3T 3H

4T 4H

5T 5H

6T 6H

You want to write the

outcomes down in an

orderly fashion.

Now calculate

So P(4 and H) = 1/12. Let’s look at it a different way.

Notice P(4 on die) = 1/6

and P(H on coin) = 1/2.

And, notice you could get 1/12 by multiplying 1/6 by 1/2. In other words, .

This is an example of a commonly used rule. We’ll state it in general but we need a definition first.

There are 12 possibilities and 1 success…

(4 ) (4) ( )P and H P P H

Independent events

Two events are independent if the occurrence of one does not affect the occurrence of the other. The events “4” and “H” are independent because the occurrence of the 4 on the die does not affect the probability that I will get an H also. The coin and die are separate items so one could not affect the other’s outcome.

Rule: Events E and F are independent if and only if the probability that they both occur is equal to the probability that E occurs times the probability that F occurs, or

.( ) ( ) ( )P E and F P E P F

Mutually exclusive events

Two events are mutually exclusive if they cannot happen at the same time.

expl: experiment: roll die

A: roll even number

B: roll a 3

Events A and B are mutually exclusive because you cannot roll an even number and a 3 at the same time.

Worksheet

“Probability Worksheet 2” covers four common types of problems. They are finding the probability that

1.) either of two mutually exclusive events occur,

2.) either of two non-mutually exclusive events occur,

3.) two independent events both occur, and

4.) two non-independent events both occur.

Its last page contains some practice problems.

“Solutions to Probability worksheet 2” is available. It explains the practice problems at the end of the worksheet.

Probability Worksheet 2There are four rules that are discussed on this worksheet. They are summarized here.

1.) If A and B are mutually exclusive, then P(A or B) = P(A) + P(B).

2.) If A and B are not mutually exclusive, then P(A or B) = P(A) + P(B) – P(A and B).

3.) If A and B are independent,then P(A and B) = P(A) P(B).

4.) If A and B are not independent, then P(A and B) = P(A) P(B given A).

Some questions

1.) What is the lowest a probability can be? Can it be 0 or negative?

2.) What is the highest a probability can be? Can it be 2?

3.) What is the sum of the probabilities of all possible mutually exclusive events?

Consider the experiment of rolling a die 100 times. What is the lowest P(4) can

be? What is the highest? What is P(1) + P(2) + P(3) + P(4) + P(5) +

P(6)?

Some answersExperimental probability is . The

number of trials will be positive. The number of successes cannot be negative but it could be zero. So, the lowest a probability can be is 0.

Since the number of successes (top) cannot exceed the number of trials (bottom), the highest a probability can be is 1.

We will investigate P(1) + P(2) + P(3) + P(4) + P(5) + P(6) on the next slide.

Die outcome

Number of times rolled

Resultant probability

1 23

2 14

3 15

4 27

5 12

6 9

23(1) 100P

14(2) 100P

15(3) 100P

27(4) 100P

12(5) 100P

9(6) 100P

Consider the data here, obtained from rolling a die 100 times.

These six events are mutually exclusive and they are the only possibilities.

What is the sum P(1) + P(2) + P(3) + P(4) + P(5) + P(6) ?

Would this be the case for any experiment and its possible

mutually exclusive events?

Complement of an eventLet A represent an event. Then (“A bar”) denotes its

complement. The complement of an event is made up of the outcomes from the sample space that are not in the original event.

Consider pulling a single card out of a deck of poker cards. Let A represent the event “red Queen”. In other words, A consists of the “Queen of Hearts” and the “Queen of Diamonds”. The complement of A would be the other 50 cards.

Since an event cannot “occur” and “not occur” at the same time, A and must be mutually exclusive. Also, since A and are the only two possibilities, the previous rule implies . We often see this written as

.

A

AA

( ) ( ) 1P A P A ( ) 1 ( )P A P A

Certain and impossible events

An event that has probability 1 must always happen. It is called a sure or certain event.

experiment: Toss coin

event: heads or tails

When you toss a coin, you must get either a heads or a tail.

An event that has probability 0 will never happen. It is called an impossible event.

experiment: Roll die

event: roll 7

When you roll a six-sided die, you cannot get a 7.

Using your sample space Consider the experiment of rolling two distinguishable six-

sided dice. Let’s find the probability that the sum of the two dice is 6. Looking at the sample space (36 events) of this experiment will help immensely.

Since there are 5 successes out of 36 equally likely possibilities, the probability of rolling a sum of 6 is .36

5

Count the number of successes and the

number of possibilities.

7.1: 1, 2, 3, 5, 7, 8, 9, 11, 12, 22, 24, 26, 27, 28, 29, 31, 34, 37b

Remember, most problems can be handled using the basic definition, number of successes divided by number of possibilities. Use the four rules discussed to do more

complicated problems.

Homework