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Math 4030-2a

Sample Space, Events, and Probabilities of Events

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Random Experiment

An experiment is called Random experiment if

1. The outcome of the experiment in not known in advance

2. All possible outcomes of the experiment are known.

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Sample space and events (Sec. 3.1)

Set of all possible outcomes of an experiment is called sample space

We will denote a sample space by S finite or infinite. discrete or continuousAny subset of a sample space is called

an event.

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Operations on events

Union, , “or” Intersections, , “and” Complement, , “not” Mutually Exclusive Events Venn diagram

A, Ac

Probability of an event (Sec. 3.3)

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Event A

P(A)P(A)SS

Axioms of probability (Sec. 3.4)

Axiom 1. 0 ≤ P(A) ≤ 1.

Axiom 2. P(S) = 1

Axiom 3. If A and B are mutually exclusive events then

P(A U B) = P(A) + P(B)

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Axioms of probability

Generalization of Axiom 3.

If A1, A2, …, An are mutually exclusive events in a sample space S then

P(A1 U A2 U … U An) =

P(A1) + P(A2) + … + P(An)

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Addition rule of probability

If A and B are any events in S then

P(AUB) = P(A) + P(B) – P(A B)

Special case: if A and B are mutually exclusive, then

P(AUB) = P(A) + P(B).

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Probability rule of the complement

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If B is the complement of A, then

P(B) = 1 - P(A).

Classical probability has assumptions:

There are m outcomes in a sample space (as the result of a random experiment);

All outcomes are equally likely to occur;An event A (of our interest) consists of s

outcomes;Then the definition of the probability for

event A is.)Pr()(

m

sAAP

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Relative frequency approach

Perform the experiment (trial) m times repeatedly;

Record the number of experiments/trials that the desired event is observed, say s;

Then the probability of the event A can be approximated by

.)Pr()(m

sAAP

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Count without counting:

Sample Space

Event

Pr(Event) = Pr(Event) =

Count!

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Multiplication Rule (P50):

• k stages;

• there are n1 outcomes at the 1st stage;

• from each outcome at ith stage, there are ni outcomes at (i+1)st stage; i=1,2,…,k-1.

Total number of outcomes at kth stage is

knnn 21

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Permutation Rule (P51):

• n distinct objects;

• take r (<= n) to form an ordered sequence;

Total number of different sequences is

121 rnnnnPrn

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Factorial notation:

12321! nnnPn nn

Permutation number when n = r, i.e.

!!

121

rn

n

rnnnnPrn

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Combination Rule (P52):

• n distinctive object;

• take r (<= n) to form a GROUP (with no required order)

Total number of different groups is

!!

!

!

121

rnr

n

r

rnnnn

r

nCrn

Count without counting:

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Multiplication: Independency between stages;

Permutation: Choose r from n (distinct letters) to make an ordered list (words). Special case of multiplication;

Factorial: Special case of permutation;

Combination: Choose r from n, with no order.