Event – set of outcomes

Venn Diagrams

Outcomes are mutually exclusive – disjoint
S
1
4 Event A
2 5
3
6
Outcomes
An Example from Card Games

What is the probability of drawing two of the same card in a row in a shuffled deck of cards?

Communications Speech and Image Processing Machine Learning Decision Making Network Systems Artificial Intelligence
Used in many undergraduate courses (every grad course)

By their average behavior By the likelihood of particular outcomes

Allows us to build models for many physical behaviors

Speech, images, traffic …
Applications
Introduction to Probability: Counting Methods
Rutgers University Discrete Mathematics for ECE 14:332:202
ห้องสมุดไป่ตู้
Why Probability?

We can describe processes for which the outcome is uncertain

Experiment

Roll a dice Roll a six 1,2,…6 Dice rolled is odd

Outcome – any possible observation of an exp.


Sample Space – the set of all possible outcomes

# outcom es _ in _ event _ space P(Event) = # outcom es _ in _ sam ple_ space

Expressed as the ratio of favorable outcomes to total outcomes

-- Only when all outcomes are EQUALLY LIKELY
Probabilities from Combinations

Rule of Product:


Total number of two card combinations? We need to find all the combinations of suit and value that describe our event set: use rule of product to find the number of combinations First, we find number of values – 13 choices, and choices of suits: 4 4!
Combinatorics



Number of ways to arrange n distinct objects n! Number of ways to obtain an ordered sequence of k objects from a set of n: n!/(n-k)! -- k permutation Number of ways to choose k objects out of n distinguishable objects:
n n! k k!(n k )!
This one comes up a lot!
Set Theory and Probability

We use the same ideas from set theory in our study of probability


Coin flipping Dice rolling Card Games
Combinatorics

Mathematical tools to help us count:
How many ways can 12 distinct objects be arranged? How many different sets of 4 objects be chosen from a group of 20 objects? -- Extend this to find probabilities …
Methods of Counting



One way of interpreting probability is by the ratio of favorable to total outcomes Means we need to be able to count both the desired and the total outcomes For illustration, we explore only the most important applications:

Event Space


Sample Space

Sample Space/Event Space

Venn Diagram
Event Space (set of favorable outcomes)
S all possible outcomes
{A,A} {K,2}
Calculating the Probability

Experiment

Pulling two cards from the deck All outcomes that describe our event: Two cards are the same All Possible Outcomes All combinations of 2 cards from a deck of 52