LEARNING TARGETS:Ø I CAN FIND THE PROBABILITY OF EVENTSØ I CAN FIND THE PROBABILITIES OF MUTUALLY EXCLUSIVE EVENTSØ I CAN FIND THE PROBABILITIES OF INDEPENDENT EVENTSØ I CAN FIND THE PROBABILITY OF THE COMPLEMENT OF AN EVENT

9.7 - Probability

The Probability of an Event

Any happening for which the result is uncertain is called an experiment.

The possible results of the experiment are outcomes.

The set of all possible outcomes of the experiment is the sample space of the experiment.

The Probability of an Event

If you flipped a nickel and then flipped a penny, there would be 4 possible outcomes in the sample space.

Example:1) Find the sample space for tossing three coins.

H T

H H T T

H H H T T TH

H H

HH TH HT

T

T T

T

T TTH H TTH HSample Space

The Probability of an Event

The number of outcomes in an event 𝐸 is denoted by 𝑛 𝐸 .

The number of outcomes in the sample space 𝑆 is denoted by 𝑛 𝑆 .

𝑃 𝐸 =𝑛 𝐸𝑛 𝑆

If an event 𝐸 has 𝑛 𝐸 equally likely outcomes and its sample space 𝑆 has 𝑛 𝑆 equally likely outcomes, the probability of an event 𝐸 is:

Example:

2) Two coins are tossed. What is the probability that both land heads up?

𝐸 = 𝐻𝐻𝐸 = 𝑆 = 𝐻𝐻, 𝑇𝐻,𝐻𝑇, 𝑇𝑇𝑆 =

𝑃 𝐸 = =14

𝑛(𝐸)𝑛(𝑆)

Example:3) Two six-sided dice are tossed. What is the probability that the total of the two dice is 7?

= 𝑛(𝑆)Total # of possibilities = 6 × 6 = 36

# of ways to roll a 7 = 𝑛(𝐸)= 6

𝑃 𝐸 =𝑛(𝐸)𝑛(𝑆)

=636 =

16

Example:

4) The numbers of colleges in various regions of the US in 2001 are shown below. One college is selected at random. What is the probability that it is in one of the 3 southern regions?

= 𝑛 𝑆 = 4178Total # of colleges

# of colleges in the 3 southern regions= 𝑛 𝐸 = 1344

𝑃 𝐸 =𝑛(𝐸)𝑛(𝑆) =

13444178 ≈ 0.322

Video Time

Mutually Exclusive Events:Two events 𝐴 and 𝐵 are mutually exclusive if 𝐴 and 𝐵 have no outcomes in common.

Examples of mutually exclusive events:

1) Choosing a student who is both a junior AND a freshman.

2) Rolling a die and getting a 2 AND an odd number.

3) Choosing a card from a standard deck of cards that’s both a red card and a spade.

Unions & Intersections:The union, ∪, of two things is all of those things put together.

In a word problem, “or” means union.

Unions & Intersections:The intersection, ∩, of two things is only what those things have in common.

In a word problem, “and” means intersection.

Probability of the Union of Two events:

If 𝐴 and 𝐵 are events in the same sample space, the probability of 𝐴or 𝐵 occurring is given by:

𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃 𝐴 ∩ 𝐵

If 𝐴 and 𝐵 are mutually exclusive, then:

𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵

Example:

5) One card is selected at random from a standard deck of 52 cards. What is the probability that the card is either a heart or a face card?

5) One card is selected at random from a standard deck of 52 cards. What is the probability that the card is either a heart or a face card?

The deck has 13 hearts, so the probability of choosing a heart is:𝑃 𝐴 =

1352

The deck has 12 face cards, so the probability of choosing a face card is:

𝑃 𝐵 =1252

Example:

5) One card is selected at random from a standard deck of 52 cards. What is the probability that the card is either a heart or a face card?

5) One card is selected at random from a standard deck of 52 cards. What is the probability that the card is either a heart or a face card?

𝑃 𝐴 ∩ 𝐵 =352

Because 3 of the cards are hearts AND face cards, the probability of choosing a heart that’s also a face card is:

So, the probability of choosing a heart OR a face card is:

𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃 𝐴 ∩ 𝐵 =1352 +

1252 −

352 =

2252 ≈ 0.423

Example:

6) The personnel department of a company has compiled data on the numbers of employees who have bene with the company for various periods of time. The results are shown in the table.

If an employee is chosen at random, what is the probability that the employee has 9 or fewer years of service?

Example:

Let’s let 𝑃(𝐴) be choosing an employee with 0 to 4 years of service. Then

First, let’s find the total number of employees = 529.

𝑃 𝐴 =157529

Let’s let 𝑃(𝐵) be choosing an employee with 5 to 9 years of service. Then

𝑃 𝐵 =89529

Example:Events 𝐴 and 𝐵 are mutually exclusive. So,

=157529 +

89529

=246529 ≈ 0.465

𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵

Independent Events:

Two events are independent if the occurrence of one has no effect on the other.

For example, rolling a 5 and picking a King.

If 𝐴 and 𝐵 are independent events, the probability that both 𝐴 and 𝐵 will occur is

𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃 𝐴 ∗ 𝑃(𝐵)

This is also called joint probability.

Example:7) A random number generator on a computer selects three integers from 1 to 20. What is the probability that all three numbers are less than or equal to 5?

The probability of selecting a number from 1 to 5 is 𝑃 𝐴 =520

=14

So, the probability that all three numbers are from1 to 5 is 14∗14∗14=164

The Complement of an Event:The complement of an event 𝐴 is the collection of all outcomes in the sample space that are not in 𝐴.

The complement of event 𝐴 is denoted by 𝐴′.

𝑃 𝐴 𝑜𝑟 𝐴D = 1 and 𝐴 and 𝐴D are mutually exclusive. So, 𝑃 𝐴D = 1 − 𝑃(𝐴)

For example, if 𝑃 𝐴 = EF

then

𝑃 𝐴′ = 1 − EF= G

F

Practice Problems:

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