4.1-4.2 Sample Spaces and Probability

Should I carry an umbrella today?

Will my car battery last until spring?

Should I accept that new job?

The chance of an event occurring.

Examples: card games, slot machines,

lotteries, …insurance, investments, weather

forecasting

Basis of inferential statistics

Probability Experiment: A chance process

that leads to well-defined results called

outcomes.

Outcome: The result of a single trial of a

probability experiment.

Trial: one flip of a coin, one roll of a die, etc.

Sample Space: The set of all possible

outcomes of a probability experiment.

Die 1

Die 2

1 2 3 4 5 6

1 (1,1)

2 (1,2)

3 (1,3)

4

5

6

Die 1

Die 2

1 2 3 4 5 6

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

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

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

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

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

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

H A 2 3 4 5 6 7 8 9 10 J Q K

D A 2 3 4 5 6 7 8 9 10 J Q K

S A 2 3 4 5 6 7 8 9 10 J Q K

C A 2 3 4 5 6 7 8 9 10 J Q K

52 Possible Outcomes

Kid 3Kid 2Kid 1

Boy

BoyBoy

Girl

GirlBoy

Girl

Kid 3Kid 2Kid 1

Girl

BoyBoy

Girl

GirlBoy

Girl

8

Possibilitie

s with

Three

Children

A Tree Diagram is a device consisting of line

segments emanating from a starting point

and also from the outcome point. It is used

to determine all possible outcomes of a

probability experiment.

An Event consists of a set of outcomes of a

probability experiment.

Simple Event: an event with one outcome

(rolling a die one time, choosing one card)

Compound Event: an event with more than

one outcome (rolling an odd number on one

die -3 possibilities)

Classical

Empirical (Relative Frequency)

Subjective

Uses sample spaces to determine numerical

probability that an event will happen.

An experiment is not performed to determine

the probability of an event.

Assumes that all outcomes in a sample

space are equally likely to occur (6

possibilities on a die have equally likely

chance of occurring)

Probability of any event E is

Number of outcomes in E .

Total number of outcomes in the sample space

This probability is denoted by

P(E) = n(E)

n(S)

Answers given as fractions, decimals or

percentages.

Reduced fractions or decimals rounded to

two or three decimal places

If probability is extremely small, round the

decimal to the first nonzero digit after the

decimal point. (0.000000478 = 0.0000005).

And means “at the same time.”

Or means

› Inclusive or (drawing a queen or a heart means

looking for one of 4 queens or one of 13 hearts.

Q of H included in both sets, so possibilities are

4 + 13 -1 = 16)

› Exclusive or (drawing a queen or a king means

looking for one of 4 queens or one of 4 kings. 4

+ 4 = 8 possibilities).

A card is drawn from a standard deck. Find

these probabilities:

› A) Of getting a 10.

› B) Of getting the 5 of clubs (a 5 and a club)

› C) Of getting a 7 or a heart

› D) Of getting an Ace or a 2

1. The probability of any event E is a number

(either a fraction or a decimal) between and

including 0 and 1. This is denoted by 0 ≤

P(E) ≤ 1.

2. If an event E cannot occur (the event

contains no members in the sample space),

its probability is 0.

3. If an event E is certain, then the probability

of E is 1.

4. The sum of the probabilities of all the

outcomes in the sample space is 1.

The Complement of event E is the set of

outcomes in the sample space that are not

included in the outcomes of event E. The

complement of E is denoted by Ē (E “Bar”).

Find the complement of selecting a letter of

the alphabet and getting a vowel.

P(Ē) = 1 – P(E) or P(E) = 1 - P(Ē) or

P(E) + P(Ē) = 1

Used to pictorally represent the probability of

events.

Venn Diagram for the probability and

complement:

P(S) = 1

P(E)

P(Ē)

P(E)

The type of probability that uses frequency distributions based on observations to determine numerical probabilities of events.

For example, one might actually roll a given die 6,000 times to observe the frequencies of each possibility. They would then use the outcomes of the experiment upon which to base their probability.

Given a frequency distribution, the

probability of an event being in a given class

is

P(E) = Frequency for class = f .

Total frequencies in the distribution n

This probability is called empirical

probability and is based on observation.

For a recent year, 51% of the families in the US had no children under the age of 18; 20% had one child; 19% had two children; 7% had three children; and 3% had four or more children. If a family is selected at random, find the probability that the family has› Two or three children

› More than one child

› Less than three children

› Based on the answers in the first three parts, which is most likely to occur?

When a probability experiment is repeated a

large number of times, the relative frequency

probability of an outcome will approach its

theoretical probability.

The type of probability that uses a probability

value based on an educated guess or

estimate, employing opinions and inexact

information.

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