W8 Probability and Counting Rules

Probability

can be defined as the chance

of an event occurring. It can be used to

quantify what the “odds” are that a specific

event will occur. Some examples of how

probability is used everyday would be

weather forecasting, “75% chance of

snow” or for setting insurance rates.

probability experiment

is a chance

process that leads to well-defined results

called outcomes.

outcome

is the result of a single trial of a

probability experiment.

sample space

is the set of all possible

outcomes of a probability experiment.

event

consists of outcomes.

Independent Events

• A occurring does NOT affect the

probability of B occurring.

“AND” means to MULTIPLY!

• FORMULA

P(A and B) = P(A) • P(B)

also known as

P(A ∩ B) = P(A) ● P(B)

Dependent Events

• A occurring AFFECTS the probability of B occurring

• Usually you will see the words  “without replacing”

“AND” still means to MULTIPLY!

Formula

P(A and B) = P(A) • P(B given A)

also known as

P(A ∩ B) = P(A) ● P(B|A)

1. Classical probability

2. Empirical probability

3. Subjective probability

Sample Spaces and Probability

There are three basic interpretations of

probability:

Classical probability

uses sample spaces to

determine the numerical probability that an

event will happen and assumes that all outcomes

in the sample space are equally likely to occur.

Rounding Rule for Probabilities

Probabilities should be expressed as reduced fractions

or rounded to two or three decimal places. When the

probability of an event is an extremely small decimal,

it is permissible to round the decimal to the first

nonzero digit after the decimal point.

Probability Rule 1

• The probability of any event E is a number (either a fraction or

decimal) between and including 0 and 1.

• This is denoted by 0 ≤ P(E) ≤ 1.

Probability Rule 2

If an event E cannot occur (i.e., the event contains no members

in the sample space), its probability is 0.

Probability Rule 3

If an event E is certain, then the probability of E is 1.

Probability Rule 4

The sum of the probabilities of all the outcomes in the sample

space is 1.

Empirical probability

relies on actual

experience to determine the likelihood of

outcomes.

Subjective probability

uses a probability value

based on an educated guess or estimate,

employing opinions and inexact information.

Addition Rules for Probability

Two events are mutually exclusive events if

they cannot occur at the same time (i.e., they

have no outcomes in common)

Multiplication Rules

■ Two events A and B are independent events if

the fact that A occurs does not affect the

probability of B occurring.

Conditional probability

is the probability that the second event B occurs given that the first event

A has occurred.

Counting Rules

■The fundamental counting rule is also called the

multiplication of choices.

■In a sequence of n events in which the first one

has k1, possibilities and the second event has k2

and the third has k3, and so forth, the total number of possibilities of the sequence will be

Factorial

is the product of all the positive numbers from 1 to a number.

Permutation

• is an arrangement of objects in a

specific order. Order matters.

Combination

• is a grouping of objects. Order

does not matter.