Introduction to Probability: Counting Methods

1. Introduction to Probability: Counting Methods Rutgers University Discrete Mathematics for ECE 14:332:202

2. Why Probability? • We can describe processes for which the outcome is uncertain • By their average behavior • By the likelihood of particular outcomes • Allows us to build models for many physical behaviors • Speech, images, traffic …

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

4. 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: • Coin flipping • Dice rolling • Card Games

5. 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 …

6. 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: This one comes up a lot!

7. Set Theory and Probability • We use the same ideas from set theory in our study of probability • Experiment • Roll a dice • Outcome – any possible observation of an exp. • Roll a six • Sample Space – the set of all possible outcomes • 1,2,…6 • Event – set of outcomes • Dice rolled is odd

8. Venn Diagrams • Outcomes are mutually exclusive – disjoint S 2 3 1 5 4 6 Event A Outcomes

9. 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? • Experiment • Pulling two cards from the deck • Event Space • All outcomes that describe our event: • Two cards are the same • Sample Space • All Possible Outcomes • All combinations of 2 cards from a deck of 52

10. Sample Space/Event Space • Venn Diagram Event Space (set of favorable outcomes) S all possible outcomes {A,A} {K,2}

11. Calculating the Probability • P(Event) = • Expressed as the ratio of favorable outcomes to total outcomes • -- Only when all outcomes are EQUALLY LIKELY

12. 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: to give our number of possible outcomes  13*6 = 78 • Probability(Event) = 78/1326 = 0.0588

13. Probabilities from Subexperiments • Only holds for independent experiments • Let’s look at the last problem: • Two subexperiments: • First can be anything 52/52 = 1 • Second, must be one of the 3 remaining cards of the same value from 51 remaining cards  3/51 = 0.588

14. An Example from Dice Rolling • Experiment: Roll Two (6-sided) Dice • Event: Numbers add to 7 • Sample Space: (all possible outcomes)S =

15. Sample Space/Event Space Event Space • Venn Diagram S

16. Calculating Probability • P(Event) = = 6/36 = 1/6

17. Side Note • Probability is something we calculate “theoretically” as a value between 0 and 1, it is not something calculated through experimentation (that is more statistics). • Just because you roll a dice 100 times, and it came up as a 1 20 times, does not make P(roll a 1) = 0.2 • It would be the limiting case in doing an infinite number of experiments, but this is impossible. • So, call your calculated values the “probability”, and your experimental values the “relative frequency”.