Discrete mathematics cs 2610
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Discrete Mathematics CS 2610. November 5, 2008. Probability Theory. When dealing with experiments for which there are multiple outcomes- x 1 , x 2 , …, x n –we require 0  p( x i )  1 for i = 1, 2, …, n and (i=1, n) p( x i ) = 1

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Discrete Mathematics CS 2610

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Discrete mathematics cs 2610

Discrete Mathematics CS 2610

November 5, 2008


Probability theory

Probability Theory

  • When dealing with experiments for which there are multiple outcomes- x1, x2, …, xn –we require

    • 0  p(xi)  1 for i = 1, 2, …, n and

    • (i=1, n) p(xi) = 1

  • We can treat p as a function that maps elements from the sample space to real values in the range [0,1]. We call such a function a probability distribution.


Probability theory1

Probability Theory

Uniform Probability Distribution:

p(xi) = 1/n, for i = 1, 2, …, n

All outcomes are equally probable.


Probability theory2

Probability Theory

Note that sum and product rules apply when dealing with probabilities too!

Sequences of events are products

Either/or requires sum rule and subtraction principle

Complementary rule works too!


Conditional probability

Conditional Probability

The conditional probability of E given F is

P(E | F) = p(E  F) / p(F)

This is the probability that E will/has occurred if we know that F has/will occur.


Independence

Independence

Two events, E and F, are independent iff

p(E1  E2) = p(E1) p(E2)

The two events don’t influence one another!


Repeated trials

Repeated trials

If there are a number of trials being conducted, each of which has a probability of success of p and a probability of failure of q = 1 – p, then the probability of exactly k successes in n independent trials is

C(n,k)pkqn-k

This is called the binomial distribution.


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