1. Probability Test Review (What are your chances of passing?)

2. See if you know the probability of: Rolling a 3 with two dice? 2 out of 36 = 1/18 th 5.6% Rolling a 9 with two dice? 4 out of 36 = 1/9 th 11.1% Rolling “doubles?” 6 out of 36 = 1/6 th 16.7% Rolling a 7 with two dice? 6 out of 36 = 1/6 th 16.7%

3. How many ways….

4. Odds vs. Probability The probability of rolling a 7 with two dice is 1/6 th. What are the odds in favor? Odds against? Odds in favor are 1 to 5. Odds against are 5 to 1. The two numbers in the odds ratio add up to the total number of ways in which an event can occur. Frequently that ratio can be simplified. Suppose there are 100 possibilities in a game, and 40 ways to win, 60 ways to lose. What are the odds in favor of winning? 40 to 60 = 4 to 6 = 2 to 3. Odds against are 3 to 2. If the probability in favor is 3/7 th then what are the odds against? 4 to 3 against. (3 ways to “win” 4 ways to lose, lose to win = 4:3)

5. Pascal’s Triangle “zero” row 1 “1” row 1 1 “2” row 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 What are the numbers in the “9” row? 1 9 36 84 126 126 84 36 9 1

6. Flipping a Coin or True-False using Pascal’s Triangle What’s true about the sum of every row in Pascal’s Triangle? They are all powers of 2: 1…2…4…8…16…32…64…128…256…512… What’s also true about each number in Pascal’s Triangle? Each one is a combination number. For example, if you have 9 items and choose any 4 at random, that’s 9 C 4 = 126 ways. How many ways could you get (by guessing alone) exactly 7 questions right on a ten-question true-false test? (See the “10” row) 10C7 = 120 ways Flip ten coins, exactly 7 come up heads? Same answer: 120 ways

7. Adding Probabilities What about “at least” getting 7 out of 10 right by guessing alone? Use the “10” row of Pascal’s Triangle: 1 10 45 120 210 252 210 120 45 10 1 Add the values for 7 out of 10 (120), 8 out of 10 (45), etc. 120 + 45 + 10 + 1 = 176 different combinations. The probability is 176 out of 1024. Simplify that fraction. 11/64 That’s just a bit above 17% (0.171875) How about getting at least half right by guessing? 638 / 1024 ≈ 62.3 %

8. Playing Cards There are 52 cards in a standard deck of cards. (without jokers) Any combination of a “hand” will most likely start with 52. How many five-card hands are there? Use 52 C 5 How many possibilities are there? 2,598,860 Wow! How many “Spades” hands? (13 cards) 52 C 13 = ?? 635,013,559,600 Boy! That’s a lot of different combos!