1. T—05/26/09—HW #71: Pg 713: 52 - 58; Pg 719: 12, 13, 24 - 29; Pg 734: 12 - 17; Pg 742: 10 – 13 52) 2170054) 379756) perm, 21058) comb, 924 12).524) e: 13/60, t: 1/626) e: 59/120 t: ½ 28) e: 29/40, t: 2/3 12).04714).05916).020 10).000019112).0148

2. W—05/27/09—HW #72: Pg 706: 47, 61; Pg 713: 47, 51; Pg 734: 4 - 9; Pg 743: 19, 21; Pg 759: 28, 29 734: 4).276).28).5 759: 28) 16

3. Chapter 12 Review

4. Given a hamburger, cheeseburger, and double-double for burgers, fries light, fries well done, regular fries, animal fries for fries, and a drink, shake, coffee, or milk for a beverage, how many different combos could you make? Fundamental Counting Principle If one event occurs in m ways, and another in n ways, the number of ways both can occur is m n ways. This can be extended to more than two events. Burgers (3 ways) Fries (4 ways) Drinks (4 ways) X X = 48 ways

5. Repetition versus non-repetition If you had to make a 5-digit ID number where the first number must be 0 or 1, how many possible ID numbers are there: A) If digits can repeat. B) If digits cannot repeat. How many options are there for the first blank? 2 How many options are there for the 2 nd blank? 10 Don’t forget 0 10 =20,000 How many options are there for the first blank? 2 How many options are there for the 2 nd blank? 9 Why? Because you can’t repeat a number you used before, so there is one less option. 8 76 =6,048

6. n! is read as “n factorial” and is represented by: (obligatory shout joke) n(n – 1)(n – 2)…321 = n! Or basically, 5! = 54321 = 120 Don’t multiply it all out, use some common sense. For factorials on your calculator, it’s “Math”  “PRB”  “!” 0! = 1  IMPORTANT

7. There are 10 people taking a test, with no one getting the same score. How many different ways can the students be ranked? How many different ways can the students finish in first or second? 10 9 8 7 6 5 4 3 2 1 = 3,628,800 10 9 = 90 This second example is special. There is a formula for an example like this: Permutations of n objects taken r at a time The number of permutations of r objects taken from a group of n distinct objects is denoted by n P r and is given by: What this is saying is if I have n things to pick from, and pick r items, how many different ways can I arrange them?

8. There are 7 food places Mr. Kim wants to visit, but he can only visit 4 of them. In how many different orders can Mr. Kim visit all these places? Calculator: press 7 first, then “math”  “PRB”  “nPr”, then 4. There are 16 players on varsity baseball, and 9 spots on a line-up, how many different ways can the coach arrange to have all the batters hit?

9. We learned that permutations is when order is important. When order is NOT important, the number of different possibilities are called COMBINATIONS. Permutation: How many different 9 man batting orders can you make with 16 players? Combinations: How many different combinations of 9 players can play if you have 16 players? (Batting order doesn’t matter) Permutation: Given 10 classes to choose from, how many 5 period schedules could you make? Combinations: Given 10 classes, how many combinations of 5 classes could you make? (The order of classes don’t matter)

10. Combinations of n objects taken r at a time The number of combinations of r objects taken from a group of n distinct objects is denoted by n C r and is given by: n  how many to choose from r  how many you pick

11. 13 topping options, how many pizzas can be made with AT MOST 2 toppings.

12. When all outcomes are equally likely, the THEORETICAL PROBABILITY that an event A will occur is: Things to know: Prime numbers are numbers where the factors are 1 and itself Perfect square are numbers with “nice” square roots, like 25 Factors are numbers that divided evenly with a number. Multiples are if you multiply a number, like multiples of 3 are 3, 6, 9, 12, etc Sum means add

13. 123456 1234567 2345678 3456789 45678910 56789 11 6789101112 Sum dice chart An odd sum A prime number The number five

14. Mutually exclusive – Nothing in common. P(A and B) = 0 Compound – Something in common

15. Fill in the blanks, say if it is mutually exclusive or not

16. Probability of drawing a heart or a face card? Probability of drawing a spade or club? Probability of drawing an even number?

17. Independent  One event has NO effect on the other. Examples: Flipping a coin. Spinning a wheel. Picking a card then putting it back. Dependent  One event HAS AN effect on the other. Examples: Picking a card and NOT putting it back. Picking a marble out of a bag and not putting it back. Check hw

18. If A and B are INDEPENDENT events, then the probability that both A and B occur is: P(A and B) = P(A) * P(B) If A and B are DEPENDENT events, then the probability that both A and B occur is: P(A and B) = P(A) * P(B|A) That line means probability of B considering A already happened. This is called the CONDITIONAL PROBABILITY OF B given A. Check hw

19. Classic Case involves Cards. Terminology: With replacement  You pick the card, you put it back, so the previous choice DOES NOT affect the next one. Without replacement  You pick the card, you DO NOT put it back, so the previous choice DOES affect the next one. W\ ReplacementW\O Replacement First card Ace Second card Face First card Heart Second card Heart Perfect vs Imperfect information Application to Hold ‘Em, I do not endorse it

20. W\ Replacement W\O Replacement Check hw

21. Coin toss Multiple Choice (SAT style) Inequalities with k, n = #, p = #

22. T—05/26/09—HW #71: Pg 713: 52 - 58; Pg 719: 12, 13, 24 - 29; Pg 734: 12 - 17; Pg 742: 10 - 13 W—05/27/09—HW #72: Pg 706: 47, 61; Pg 713: 47, 51; Pg 734: 4 - 9; Pg 743: 19, 21; Pg 759: 28, 29