Math 409/409GHistory of Mathematics

The Probability of

Simple Events

What is probability?

Probability is the measure of the likelihood of a random phenomenon or chance behavior.

Definitions

A simple event is any single outcome from a probability experiment.

A sample space, S, of a probability experiment is the collection of all simple events.

An event is any collection of outcomes from a probability experiment. An event consist of one or more simple events.

Example

A probability experiment consists of rolling a single “fair” die.

• What are the simple events of this probability experiment?

• What is the sample space?

• Give two examples of events that are not simple events.

Answers• The simple events are the possible

outcomes of rolling the die. Since there are 6 numbers on the die, the simple events are:

“Rolling a 1” {1} “Rolling a 2” {2} “Rolling a 3” {3} “Rolling a 4” {4} “Rolling a 5” {5} “Rolling a 6” {6}

• The sample space, S, is the set of all simple events. So

S {1, 2, 3, 4, 5, 6}

• An event consists of one or more simple events. So two non-simple events are:

E {2, 4, 6} “Rolling an even number”

O {1, 3, 5} “Rolling an odd number”

Another Definition

• A probability experiment is said to have equally likely outcomes if each simple event in the sample space has the same chance (probability) of occurring.

Example: When rolling a “fair” die, each number on the die has the same chance of occurring as any other number. But if you replace the number 6 on the die with the number 2, then you no longer have equally likely outcomes since rolling a 2 has a better chance or occurring.

Computing Probabilities

If an experiment has n equally likely simple events and if the number of ways that event E can occur is m, then the probability of event E is

number of ways that can occurPr [ ]

total number of possible outcomes

EE

m

n

Examples

A probability experiment consists of rolling a single “fair” die. Then:

1

6Pr [rolling a 2]

3 1

6 2Pr [rolling an even number]

4 1

6 3Pr [rolling a no. larger than 2]

01 0

6Pr [rolling a ]

Important Properties

If S is the sample space and E is an event in that sample space, then:

1Pr [ ] S

0 1Pr [ ] E

Mutually Exclusive Events

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

Example: When rolling a single fair die, the events of rolling a 1 and of rolling a 2 are mutually exclusive events since the die will show only one number. But the events of rolling a 1 and of rolling an odd number are not mutually exclusive since 1 is an odd number.

Addition Rules

• If A and B are mutually exclusive events, then

• If A and B are not mutually exclusive events, then

Pr[ or ] Pr[ ] Pr[ ]A B A B

Pr[ or ] Pr[ ] Pr[ ] Pr[ and ]A B A B A B

Examples

• When rolling a single fair die, what’s the probability of rolling a 1 or a 2?

1 1 10

6 6 3

Pr[1 or 2] Pr[1] Pr[2] Pr[1 and 2]

• When rolling a fair die, what is the probability of rolling a 1 or rolling an odd number?

First note that event of rolling a 1 and rolling an odd number is the same as the event of rolling a 1.

1 3 1 1

6 6 6 2

Pr[1 or odd] Pr[1] Pr[odd] Pr[1 and odd]

Independent Events

• Two events A and B are independent if knowing whether A occurs does not change the probability that B occurs.

Example: Two marbles are drawn one at a time from an urn containing 3 red marbles and 2 blue marbles. Are the events of first drawing a red marble and then drawing another red marble independent events?

Well, that depends on whether or not the first marble is placed back in the urn.

• If the marble is put back in the urn after drawing the first marble, then the event of first drawing a red marble and then drawing another red marble are independent events since after putting the marble back, the sample space remains the same. So the probability of drawing the second red marble is the same as the probability of drawing the first red marble.

• But if the first marble is not put back, then after the first marble is drawn, the sample space has been reduced by one marble. So the probability of the second marble has changed after drawing the first marble. So the events of first drawing a red marble and then drawing another red marble are not independent events.

Multiplication Rules

• If A and B are independent events, then

• If A and B are not independent events,

where

Pr[ and ] Pr[ ] Pr[ ]A B A B

|Pr[ and ] Pr[ ] Pr[ ]A B A B A

|Pr[ ] probability of given

that has occured.

B A B

A

Examples

• Two marbles are drawn from an urn containing 3 red marbles and 2 blue marbles. What’s the probability of drawing two red marbles if the first drawn marble is placed back in the urn?

|

3 3 9

5 5 25

Pr[ and ] Pr[ ] Pr[ ]R R R R R

• Two marbles are drawn from an urn containing 3 red marbles and 2 blue marbles. What’s the probability of drawing two red marbles if the first drawn marble is not placed back in the urn?

|

3 2 6 3

5 4 20 10

Pr[ and ] Pr[ ] Pr[ ]R R R R R

The Complement of an Event

The complement, Ec, of event E is the event that E does not occur.

Example: If E is the event of rolling an even number on a fair die, then Ec is the event of rolling an even number.

Complement Rule

• For any event A, the probability that A does not occur is

Sometimes it is easier to use the complement of an event than it is to use the actual event. In this situation you want to use

1 ].Pr[ ] Pr[cA A

1 ].Pr[ ] Pr[ cA A

Example

• If you randomly select a number between (and including) 1 and 100, what is the probability that that number is less than 99?

If E is the event in this problem, it would be easier to look at Ec, the event of selecting 99 or 100. So in this case,

2 98 491 ] 1 .

100 100 50Pr[ ] Pr[ cE E

This ends the lesson on

The Probability of

Simple Events