1. Counting Introduction to Probability & Statistics Counting

2. Fundamental Rule u If an action can be performed in m ways and another action can be performed in n ways, then both actions can be performed in m n ways.

3. Fundamental Rule u Ex: A lottery game selects 3 numbers between 1 and 5 where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there?

4. Fundamental Rule u Ex: A lottery game selects 3 numbers between 1 and 5 where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there? 1234512345

5. Fundamental Rule u Ex: A lottery game selects 3 numbers between 1 and 5 where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there? 1234512345 23452345

6. Fundamental Rule u Ex: A lottery game selects 3 numbers between 1 and 5 where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there? 1234512345 23452345 345345

7. Fundamental Rule u Ex: A lottery game selects 3 numbers between 1 and 5 where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there? 1234512345 23452345 345345 LN = 543 = 60

8. Combinations u Suppose we flip a coin 3 times, how many ways are there to get 2 heads?

9. Combinations u Suppose we flip a coin 3 times, how many ways are there to get 2 heads? Soln: List all possibilities: H,H,HH,T,T H,H,TH,T,H H,T,HT,H,H T,H,HT,T,T

10. Combinations Of 8 possible outcomes, 3 meet criteria H,H,HH,T,T H,H,TH,T,H H,T,HT,H,H T,H,HT,T,T

11. Combinations If we don’t care in which order these 3 occur H,H,T H,T,H T,H,H Then we can count by combination.

12. Combinations u Combinations n C k = the number of ways to count k items out n total items order not important. n = total number of items k = number of items pertaining to event A

13. Example u How many ways can we select a 4 person committee from 10 students available?

14. Example u How many ways can we select a 4 person committee from 10 students available? No. Possible Committees =

15. Example u We have 20 students, 8 of whom are female and 12 of whom are male. How many committees of 5 students can be formed if we require 2 female and 3 male?

16. Example u We have 20 students, 8 of whom are female and 12 of whom are male. How many committees of 5 students can be formed if we require 2 female and 3 male? Soln: Compute how many 2 member female committees we can have and how many 3 member male committees. Each female committee can be combined with each male committee.

17. Example

18. Permutations u Permutations is somewhat like combinations except that order is important.

19. Example u How many ways can a four member committee be formed from 10 students if the first is President, second selected is Vice President, 3rd is secretary and 4th is treasurer?

20. Example u How many ways can a four member committee be formed from 10 students if the first is President, second selected is Vice President, 3rd is secretary and 4th is treasurer?

21. Example u How many ways can a four member committee be formed from 10 students if the first is President, second selected is Vice President, 3rd is secretary and 4th is treasurer? 10 P 4 = 10*9*8*7 = 5,040