MAT 119 FALL 2001

MAT 119

FINITE MATHEMATICS

NOTES

PART 2 – PROBABILITY

CHAPTER 7

PROBABILITY

7.1Sample Spaces and the Assignment of Probabilities

Sample space, S – includes all outcomes (element) that can occur in a real or conceptual experiment

Eg;The experiment of tossing a coin once

S = {H, T}

The experiment of tossing a coin once

S = {HH, HT, TH, TT}

Constructing a Probability Model

1.Find the sample space. List all outcomes, if too many, find the number of outcomes.

2.Assign each e a probability P(e) so that

a.P(e)  0

b.sum of probabilities assigned to all outcomes is 1.

Event – any subset of the sample space (2 or more outcomes)

simple event – has only one outcome

Probability of an Event

If E = Ø, the event is impossible and P(Ø) = 0

If E = {e} is a simple event, P(E) = P(e) is the probability assigned to outcome e.

If E is the union of r simple events {e1}, {e2}, …, {er} and

P(E) = P(e1) + P(e2) + … + P(er)

7.2Properties of the Probability of an Event

Mutually exclusive events (2 or more from s sample space) – have no common outcomes

Let E and F be mutually exclusive events,

E  F = 

P(E or F) = P(E  F) = P(E) + P(F) since P(E  F) = 0

Additive Rule

For any two events A and B of a sample space S

P(E  F) = P(E) + P(F) – P(E  F)

Properties of an Event

1.0 P(E) 1

2.P() = 0 and P(S) = 1

3.P(E  F) = P(E) + P(F) – P(E  F)

Complement of E,

P() = 1 – P(E)

Let a sample space S be given by S = {e1, e2, …, en} where all outcomes are equally likely.

For an event E containing m outcomes, P(E) = m/n

Probability of an event E in a sample space with equally likely outcomes

If the sample space S of an experiment has n equally likely outcomes, and the event E in S occurs m times, then

If the odds for E are a to b, then

If the odds against E are a to b, then

7.3Probability problems using counting techniques.

7.4Conditional Probability

Let E and F be events of a sample space S and suppose P(F) > 0. The conditional probability of the event E, assuming the event F, denoted by P(E/F), is defined as

Product Rule - P(E F) = P(E) . P(E/F)

7.5Independent Events

E is independent of F, if and only if, P(E/F) = P(E)

Thm.

Let E and F be events where P(E) > 0 and P(F) > 0. If E is independent of F, then F is independent of E.

If two events E and F have positive probabilities and if event E is independent of F, then F is also independent of E. E and F are called independent events.

Test for Independence

Two events E and F of a sample space S are independent events if and only if

P(E  F) = P(E) . P(F)

In words, the probability of E and F is equal to the product of the probability of E and the probability of F.

A set E1, E2, …, En of n events is called independent if the occurrence of one or more of them does not change the probability of any of the others. It can be shown that, for such events,

P(E1 E2 …  En) = P(E1) . P(E2) . … . P(En)

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DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS