Ordering a deck of cards... Lecture 3: Binomial Distribution. Example. Permutations & Combinations

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1 Ordering a dec of cards... Lecture 3: Binomial Distribution Sta 111 Colin Rundel May 16, 2014 If you have ever shuffled a dec of cards you have done something no one else has ever done before or will ever do again... There are aroximately (52!) ossible configurations of a dec of 52 to cards To ut that in context: Cells in the human body (10 14 ) Seconds since the big bag (10 18 ) Grains of sand on all beaches on earch ( ) Stars in the universe (10 23 ) Atoms in the observable universe (10 80 ) A Googol ( ) Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Examle Imagine you have a bag with 6 slis of aer numbered 1 to 6. How many different airs can you draw if you samle without relacement? Permutations & Combinations If we have n items and want to select of them without relacement, then there are j ossible outcomes. Permutations - when we care about the order in which ull out the items: j = n! (n )! Combination - when we do not care about the order in which ull out the items: ( ) n n! j = = (n )!! Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23

2 Permutations & Combinations - Derivation Some roerties of the Binomial coefficient ( ) n = n =0 ( ) ( ) n n = n ( ) n = 2 n ( ) n ( n 1 ), for 0 < < n Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Pascal s Triangle and the Binomial coefficient ( 0 0) ( 1 ) ( 1 0 1) ( 2 ) ( 2 ) ( ) ( 3 ) ( 3 ) ( 3 ) ( ) ( ) Probability Distributions Descrition of the robability for all values of a random variable (a numeric value that deends in some way on chance). We have to distinguish between the discrete and continuous case: Discrete (integer valued) - easy to assign robability to each event (even if there are infinitely many) Value of the role of a six sided die Number of coin flis until the first head Continuous (real valued) - robability defined based on an interval Probability a student s height is exactly 5 9 Probability a students height is between 5 9 and 5 10 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23

3 Distributions Functions If X taes countably-many values, then we can list them and just reort f (x) = P(X = x) When X is discrete - Probability mass function P(X A) = f (x) x A When X is continuous - Probability density function P(X A) = f (x)dx A Probability mass function Let X be the number of aces in two draws without relacement from a 52-card dec. The PMF can be resented in a variety of ways: x f(x) No aces 0 (48/52)(47/51) = One ace 1 2(4/52)(48/51) = Two aces 2 (4/52)(3/51) = Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Probability mass function Let X be the number of aces in two draws without relacement from a 52-card dec. The PMF can be resented in a variety of ways: Probability mass function Let X be the number of aces in two draws without relacement from a 52-card dec. The PMF can be resented in a variety of ways: f(x) For x {0, 1, 2}, P(X = x) = ( 4 48 ) x)( 2 x 4! 48! ( 52 ) = x!(4 x)! (2 x)!(46 + x)! x!(4 x)!(2 x)!(46 + x)! 2! 50! 52! x Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23

4 Examle Binomial Distribution A comany is testing a new manufacturing rocess for aluminum cases, if 20% of the cases do not meet their secifications what is the robability that if the comany checs the next four cases that only one of them will not meet secification? 1 Let the robability a test succeeds be = 0.8 and the robability a test fails be = 0.2 then P(1 failure in 4 tests) = = 4 3 ( ) 4 = Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Binomial Distribution We define a random variable X that reflects the number of successes in a fixed number of indeendent trials with the same robability of success as having a binomial distribution. If there are n trials then X Binom(n, ) ( ) n f ( n, ) = P(X = n, ) = (1 ) n Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23

5 What is the most robable outcome? The binomial distribution is unimodal which maes our life easier... We can loo at the ratio of successive outcomes, r = P(X = + 1), for 0 n 1 P(X = ) r is largest when = 0 and gets rogressively smaller. When r > 1 then P(X = + 1) > P(X = ) When r < 1 then P(X = + 1) < P(X = ) Maximum (mode) of the distribution occurs when r switches from being greater than 1 to less than 1. Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 What is the most robable outcome? cont. What value of results in r 1? What is the most robable outcome? cont. Max robability is therefore the smallest integer value of n. We can narrow that relationshi down somewhat since there must be an integer value of between n n + 1 n n + ( + ) n n + Secial case when r = 1 as it imlies that P(X = ) = P(X = + 1) in which case both values are eually robable. Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23

6 What is the scale of this maximum robability? Not very large... Some examles... Let X Binom(25, 0.15) then the distribution of X loos lie P(X = ) maxes out at a little bit less than 1/ n, therefore P(X = mode ) 0 as n. Concetually, as the number of bins increases the mass in each bin must necessarily get smaller, we are in essense moving from discrete to continuous distribution mode = (n, n + ) = (2.9, 3.9) = 3 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Some examles... Let X Binom(10, 6/11) then the distribution of X loos lie Outcome Ranges Often it is more interesting to tal about the robability of a range of outcomes mode = (n, n + ) = (5, 6) = 5, 6 For examle, going bac to the manufacturing examle (where =0.8, n=4) what is the robability that there are 1 or fewer defective cases? P(1 or fewer defective) = P(3 or more successes) = P(X = 3 or 4) = P(X = 3) + P(X = 4) ( ) ( ) 4 4 = (0.8) 3 (0.2) 1 + (0.8) 4 (0.2) = 4(0.1024) + 1(0.4096) = Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23

7 What haens for large n? Let x = z + n and c = P(n) then, log P(n + z) log P(n) z2 2n (x n)2 log P(x) log P(n) 2n ( P(x) ex log P(n) 1 (x n) 2 ) 2 n P(x) c e 1 (x n) 2 2 n de Moivre-Lalace Limit Theorem When n is large enough the Binomial distribution will always have this bell-curve shae. Shae of the curve given by c e b(x a)2 de Moivre and Lalace where the first to identify this attern and characterize the shae of the curve (by finding a, b, c). This is a secial case of a more general result nown as the Central Limit Theorem. (More on this later) Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23 Sta 111 (Colin Rundel) Lecture 3: Binomial Distribution May 16, / 23