Understanding Probabilities in Discrete Math: Theory & Binomial Distribution, Slides of Discrete Mathematics

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Introduction to Discrete

Mathematics

Probability Theory:

Counting in Terms of

Proportions

X

X-1 X+

X-2 X X+

X-3 X-1 X+1 X+

X-4 X-2 X X+2 X+

In the n th^ generation there will be 2n^ males, each with one of n+1 different heights: h 0 , h 1 ,…,h (^) n

h (^) i = (X-n+2i) occurs with proportion:

n i / 2^

n

Unbiased Binomial Distribution

On n+1 Elements

Let S be any set {h 0 , h 1 , …, h (^) n } where each element h i has an associated probability

Any such distribution is called an Unbiased Binomial Distribution or an Unbiased Bernoulli Distribution

n i

2 n

Teams A and B are equally good

In any one game, each is equally likely to win

What is most likely length of a “best of 7” series?

Flip coins until either 4 heads or 4 tails

Is this more likely to take 6 or 7 flips?

6 and 7 Are Equally Likely

To reach either one, after 5 games, it must be 3 to 2

½ chance it ends 4 to 2; ½ chance it doesn’t

3 choices of bag

2 ways to order bag contents

6 equally likely paths

Given that we see a gold, 2/3 of remaining paths have gold in them!

Language of Probability

The formal language of

probability is a very

important tool in

describing and

analyzing probability

distribution

Finite Probability Distribution

A (finite) probability distribution D is a finite set S of elements, where each element x in S has a non-negative real weight, proportion, or probability p(x)

Σ p(x) = 1

x ∈ S

For convenience we will define D(x) = p(x)

S is often called the sample space and elements x in S are called samples

The weights must satisfy:

Events

Any set E ⊆ S is called an event

Σ p(x)

x ∈ E

PrD [E] = S

0

PrD [E] = 0.

Uniform Distribution

If each element has equal probability, the distribution is said to be uniform

Σ p(x) =

x ∈ E

PrD [E] =

|E|

|S|

The sample space S is the

set of all outcomes {H,T}^100

Each sequence in S is

equally likely, and hence has

probability 1/|S|=1/2 100

Using the Language

S = all sequences

of 100 tosses

x = HHTTT……TH

p(x) = 1/|S|

Visually