1. Bernoulli Trials, Geometric and Binomial Probability models

2. Bernoulli Trials Three Conditions: 1. Only two outcomes (success/failure) 2. Probability of success, p, is constant for all trials 3. Each trial is independent.* *10% Condition: If we have a finite system then we can assume independence as long as we are using less than 10% of the population.

3. Example Flipping a coin and counting heads ◦ 2 outcomes: Heads, Tails ◦ p = 0.5 and is constant for each flip of the coin ◦ Each flip is independent of the last flip BERNOULLI TRIAL!!

4. Example Suppose in a game you need to roll a single standard die until you get a 6. How many rolls should it take? Bernoulli? ◦ 2 outcomes: roll a 6; don’t roll a 6 ◦ p = 1/6 that you roll a 6 and stays the same ◦ Each roll is independent!

5. Probability Model X = # of rolls 123456 P(X = x)

6. Geometric Probability Model Check that you have a Bernoulli trial first! And I mean write down each condition! Need two things p = prob of success and X = the number of trials until success Geom(p) is the model and q = 1 - p P(X = x) = q x-1 p

7. The probability that it will take the 4 th roll to get a 6 then becomes: P(X = 4) = (5/6) 4-1 (1/6) = 0.096 The probability that you will get a 6 within the first 4 rolls is: P(X≤4) = P(X=1) + P(X=2) + P(X=3) + P(X=4) = (5/6) 1-1 (1/6)+ (5/6) 2-1 (1/6)+ (5/6) 3-1 (1/6)+ (5/6) 4-1 (1/6) = 0.518

8. Expected Value of a Geometric Model Expected number of trials until success. Standard Deviation

9. How many rolls do we expect until we get a 6?

10. New Game Suppose now you play a game with 3 dice. For each 1 you roll you get 100 points What are the probabilities of the different points possible. Bernoulli? ◦ 2 outcomes: 1 or not 1 ◦ p = 1/6; is constant ◦ Each trial is independent

11. X = # of 1’s rolled 0123 P(X = x) What are all the possible combinations of rolling a die 3 times.

12. Combination If we want to know how many ways we can get k successes out of n trials we use the choose function:

13. Binomial probability model Must be Bernoulli Trials Needs two parameters: ◦ n = number of trials ◦ p = probability of success Binom(n,p) X = number of successes in n trials

14. What is the probability that you get 2 1’s out of 3 rolls? What is the probability that you get at least 2 1’s?

15. Expected Value Standard Deviation