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Tree diagrams and the binomial distribution Outline for today Better know a player: Dock Ellis Questions about worksheet 6? Review of concepts in probability: additive and multiplicative rules Tree diagrams and the analysis of Strat-o-matic The binomial distribution Announcement: class final projects Start thinking about your class final project Ideally focus on research question • Doesn’t have to be about baseball, but need to find data set that you can use to answer the question 1-2 paragraph proposal is due the Wednesday after spring break (March 23rd) Projects presentation are on April 25st Better know a player: Dock Ellis Probability Probability is a way of measuring the uncertainty of the outcome of an event Definitions: Sample space • All possible outcomes An Event • Subset of the sample space Probability key properties: • 0 ≤ Pr(X) ≤ 1 • Σ Pr(X = x) = 1 Probability rules - Additive rule If there are two events A, and B, then the probability of A or B happening is: Pr(A or B) = Pr(A) + Pr(B) – Pr(A, B) Events are called mutually exclusive if events A and B can not both occur - i.e., Pr(A, B) = 0 Q: What would mutually exclusive events look like in the Venn diagram? A: The circles would not overlap Multiplicative Rule Pr(A, B) = Pr(A|B) × Pr(B) Probability of B happening × Probability of A happening given B happened Two events are independent if: Pr(A, B) = Pr(A) x Pr(B) i.e., if the occurrence of B does not effect the probability of A happening Lottery problem Question 4c: Suppose there was a lottery where you needed to select 5 numbers from 1 to 69 but the order of the numbers mattered • E.g., (2, 4, 6, 8, 10) different entry than (4, 2, 6, 8, 10) What would the probability of winning this lottery be? 1/69 · 1/68 · 1/67 · 1/66 · 1/65 = 7.1 · 10-10 Big League Baseball For any one pitch (not assuming that the play is in play), what is the probability of a the following events? • • • • A home run? A out? A single? A Hit? 1/36 · 1/3 24/36 · 1/3 7/36 · 1/3 10/36 · 1/3 1st Die 2nd Die 1 2 3 4 5 6 1 Single Out Out Out Out Error 2 Out Double Single Out Single Out 3 Out Single Triple Out Out Out 4 Out Out Out Out Out Out 5 Out Single Out Out Out Single 6 Error Out Out Out Single Home run Strat-o-matic Much more complex board games • Takes into account Hitters and Pitchers • Advanced version accounts for additional factors (e.g., ball parks etc.) Strat-o-matic rules and tree diagrams Each player is represented by a card 1. A white single die is rolled to determine whether to use hitter or pitcher’s card: • 1-3 -> hitters card • 4-6 -> pitcher’s card 2. Then, two dice are rolled and their sum determines which play in the card should be used 3. For some plays additionally a 20 sided die is rolled to determine the final outcome • and other tables/rules often need to be consulted Strat-o-matic: analysis Let’s calculate the probability of different events… Calculating the probability of getting a particular column pitcher or hitter’s card is pretty simple • Answer? Calculating the sum of the two dice is a little more involved… • What is the sample space here? • i.e., how many possible outcomes are there? • Can you calculate the probability distribution? Strat-o-matic: analysis Fill in the table below with the sum of the two dice and then calculate the probability of rolling a 2 to a 12 2nd Die 1st Die 1 1 2 3 4 5 6 2 3 7 4 5 6 Strat-o-matic: analysis Fill in the table below with the sum of the two dice and then calculate the probability of rolling a 2 to a 12 1st Die 2nd Die 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 Strat-o-matic: analysis 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 1st Die 2nd Die 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 12 1/36 Strat-o-matic: analysis 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36 What is the probability of rolling a 1 on the white die and then getting a sum of 7 on the two red dice? • 1/6 · 6/36 = 6/216 What is the probability of rolling a 2 on the white die and then getting a sum of 8 on the two red dice, and then a number for 1-8 on the 20 sided die? • 1/6 · 5/36 · 8/20 = .00926 Strat-o-matic: analysis 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36 What is the probability of rolling a 1 on the white die or a 6 on the white die? • 1/6 + 1/6 = 2/6 What is the probability of rolling a 5 on the white die and then getting a sum of 8 or a sum of 10 on the two red dice? • 1/6 · (4/36 + 3/36)= .0324 Strat-o-matic: analysis 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36 What is the probability of rolling a 3 on the white die • and then getting a sum of 8 two red dice • or a sum of 10 on the two red dice • and then a 1-10 on the 20 sided die? Tree diagram! 5/36 8 1/6 · 5/36 = .0231 .0231 + .0069 = .0300 1/6 3 10 3/36 1-10 10/20 1/6 · 3/36 · ½ = .0069 Strat-o-matic: Pujols vs. Kershaw What is the probability of a hitting a double? Strat-o-matic: Pujols vs. Kershaw What is the probability of a hitting a double? Strat-o-matic: Pujols vs. Kershaw 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36 Tree diagram 6 5/36 1/6 1 1/6 6 14/20 5/36 6 1-14 Tree diagram 3/20 6 5/36 18-28 6/36 1/6 7 1 5/36 2/20 8 1-2 1/6 6 14/20 5/36 6 1-14 Strat-o-matic: Pujols vs. Kershaw What is the probability of a hitting a double? Tree diagram! 1/6 · [5/36 · 3/20 + 6/36 + 5/36 · 2/20] + 1/6 · (5/36 14/20) = .04977 Strat-o-matic: Pujols vs. Kershaw What is the probability of a hitting a home run? Strat-o-matic: Pujols vs. Kershaw What is the probability of a hitting a home run? Tree diagram! 1/6 · [3/36 + 4/36 + 5/36 · 17/20] + 1/6 · (4/36 · 2/20) = .0594 Back to the lottery problem Question 4c: Suppose there was a lottery where you needed to select 5 numbers from 1 to 69 but the order of the numbers mattered • E.g., (2, 4, 6, 8, 10) different entry than (4, 2, 6, 8, 10) What would the probability of winning this lottery be? 1/69 · 1/68 · 1/67 · 1/66 · 1/65 = 7.1 · 10-10 Lottery problem Question 4d: Suppose there was a lottery where you needed to select 5 numbers from 1 to 69 and the order of the numbers does not mattered • E.g., (2, 4, 6, 8, 10) is the same as (4, 2, 6, 8, 10) Solution: we need to count how many permutations there are of 5 numbers, and then use the additive rule to add all these together • i.e., multiply 7.1 · 10-10 by the number of permutations Q: How many permutations are there? Lottery problem A slightly easier problem: suppose there were 5 letters and we needed to find all the ways to select 2 letters • i.e., we have A, B, C, D, E to put into two slots There would be 5 letter choices for the 1st slot And there would be 4 letter choices for the 2nd slot A B C D E B C D E A 1 2 5 · 4 = 20 ways to permute 2 out of 5 objects Let’s write out these 20 permutations! Lottery problem Continuing this process there would be 3 letters to choose for the second slot 5 · 4 · 3 = 60 ways to permute 3 out of 5 objects Continuing this pattern we get: 5 · 4 · 3 · 2 · 1 = 120 ways to permute 5 out of 5 objects B C D E B C D E A C 1 2 B D E 3 i.e., we have factorial 5! Lottery problem Question 4d: Suppose there was a lottery where you needed to select 5 numbers from 1 to 69 and the order of the numbers does not mattered • E.g., (2, 4, 6, 8, 10) is the same as (4, 2, 6, 8, 10) Solution: we need to use the additive rules to count how many permutations there are of 5 numbers and then use the additive rule to add all these together • i.e., multiply 7.1 · 10-10 by the number of permutations A: 120 · 7.1 · 10-10 = 8.9 · 10-8 http://www.webmath.com/lottery.html n choose k We can write this as • Pr(winning lottery) = 120 · (1/69 · 1/68 · 1/67 · 1/66 · 1/65) • Or 1 in (69 · 68 · 67 · 66 · 65)/120 chance of winning More generally let’s say were are choosing k items out of n total • n = 69 • k=5 We can then calculate this using the “n choose k” function n choose k Q: How many ways are there to choose 3 things out of a total of 10? A: (10 · 9 · 8)/ (3 · 2) = 120 R: choose(n, k)