1. Today Today: Reading: –Read Chapter 1 by next Tuesday –Suggested problems (not to be handed in): 1.1, 1.2, 1.8, 1.10, 1.16, 1.20, 1.24, 1.28

2. Assignment 1

3. Combinatorics (Section 1.4) In the equally likely case, computing probabilities involves counting the number of outcomes in an event This can be hard…really Combinatorics is a branch of mathematics which develops efficient counting methods

4. Combinatorics Consider the rhyme As I was going to St. Ives I met a man with seven wives Every wife had seven sacks Every sack had seven cats Every cat had seven kits Kits, cats, sacks and wives How many were going to St. Ives? Answer:

5. Example In three tosses of a coin, how many outcomes are there?

6. Multiplication Principle Let an experiment E be comprised of smaller experiments E 1,E 2, …, E k, where E i has n i outcomes The number of outcome sequences in E is (n 1 n 2 n 3 …n k ) Example (St. Ives re-visited)

7. Example In three tosses of a coin, how many outcomes are there?

8. Tree Diagram

9. Example In a certain state, automobile license plates list three letters (A-Z) followed by four digits (0-9) How many possible license plates are there?

10. Example Suppose have a standard deck of 52 playing cards (4 suits, with 13 cards per suit) Suppose you are going to draw 5 cards, one at a time, with replacement (with replacement means you look at the card and put it back in the deck) How many sequences can we observe

11. Permutations In previous examples, the sample space for E i does not depend on the outcome from the previous step or sub-experiment The multiplication principle applies only if the number of outcomes for E i is the same for each outcome of E i-1 That is, the outcomes might change change depending on the previous step, but the number of outcomes remains the same

12. Permutations When selecting object, one at a time, from a group of N objects, the number of possible sequences is: The is called the number of permutations of N things taken n at a time Sometimes denoted N P n Can be viewed as number of ways to select N things taken n at a time where the order matters

13. Example Suppose have a standard deck of 52 playing cards (4 suits, with 13 cards per suit) Suppose you are going to draw 5 cards, one at a time, without replacement How many permutations can we observe

14. Counting Patterns Consider the word minimum How many permutations of the letters are there? How many distinguishable ways are there to to arrange these letters?

15. Counting Patterns The number of distinct sequences of N objects where m 1 are are of type 1, m 2 are are of type 2, …, m k are are of type k is: Note: N= m 1 + m 2 + …+ m k

16. Counting Patterns Consider the word minimum How many permutations of the letters are there? How many distinguishable ways are there to to arrange these letters?

17. Example In a certain state, automobile license plates list three letters (A-Z) followed by four digits (0-9) How many possible license plates are there with 7 distinct characters?

18. Combinations If one is not concerned with the order in which things occur, then a set of n objects from a set with N objects is called a combination Example –Suppose have 6 people,3 of whom are to be selected at random for a committee –The order in which they are selected is not important –How many distinct committees are there?

19. Combinations The number of distinct combinations of n objects selected from N objects is: “N choose n” Note: N!=N(N-1)(N-2)…1 Note: 0!=1 Can be viewed as number of ways to select N things taken n at a time where the order does not matter

20. Combinations Example –Suppose have 6 people, 3 of whom are to be selected at random for a committee –The order in which they are selected is not important –How many distinct committees are there?

21. Example A committee of size three is to be selected from a group of 4 Democrats, 3 Independents and 2 Republicans How many committees have a member from each group? What is the probability that there is a member from each group on the committee?