The Binomial Distribution, Mean, and Standard Deviation IBHL/SL - Santowski

(A) Review - Assumptions Recall the various key points that we have previously made for binomial probabilities: (i) we have a fixed number of trials (ii) the probability for the occurrence of our event (or success) is the same each time (iii) each trial is independent of all the other events (iv) each event only has two possible outcomes occurrence or non-occurrence (success of failure)

(A) Review - Equations Let p equal the probability of “ success” or simply our given event happening Then let q equal the probability of “ failure” or “ not successful” or simply our event not happening As well, we will run the “ experiment” n times and we will look for “ success” or the occurrence of our event in r of these n trials therefore, the “ failure, non- success ” or simply the non-occurrence of our event will happen n - r times Then the probability that the event occurs r times (or is successful r times) AND that our event does NOT occur (failure) n - r times is:  P(E=r) = C(n,r) x p r x q n-r Where P(E=r) is read as “the probability that the event occurs r times” which other texts will write as P(X = r)

(B) Examples of Working With Binomial Probabilites When working with BPD, we have a number of options available: (i) use the formula (ii) use FCP and combinatorials (iii) use a GDC and the binompdf( command ex 1  Seventy-two percent of union members are in favour of a given change in their working conditions. A random sampling of 5 members is taken. Find the probability that: (i) three members are in favour of change(ANS: ) (ii) at least 3 members are in favour of change(ANS: )

(C) Classwork/Homework SL Math textbook, Chap 29E, p723, Q1-9 HL Math textbook, Chap 30F, p742, Q1-5

(D) Mean of Binomial Probability Distributions the mean we have seen before, when we discussed the expected value and we generalized a formula that μ = np, where n is the number of events (or we have n repetitions of a binomial experiment) and p is the probability of a particular variable occurring ex 1  If we toss a coin 20 times, and the probability of getting tails is 2, then we expect to get a tail in (20)( 2 ) or 10 times ex 2  If we roll a die 48 times, and the probability of NOT getting a 3 is 5/6, then we expect NOT to get a 3 in (48)(5/6) = 40 times

(E) Standard Deviations of Binomial Probability Distributions it is a bit harder to develop a general formula for the standard deviation recall that μ = Σx i p i and that σ 2 = Σ(x i 2 p i ) - (μ 2 ) so if we run 1 trial, we could have 0 or 1 ” success” then we have μ = (0)(q) + (1)(p) = p our σ 2 = [(0) 2 q + (1) 2 (p)] - p 2 = p - p 2 = p(1 - p) = pq now if we run two trials, then we could get 0,1,2 “successes” ==> (p + q) 2 then we have μ = (0)(q)(q) + (1)(2)(p)(q) + (2)(p)(p) = 2pq + 2p 2 = 2p(q + p) = 2p(1) = 2p our σ 2 = [(0) 2 (q 2 ) + (1) 2 (2pq) + (2) 2 (p 2 )] - (2p) 2 = 2pq + 4p 2 - 4p 2 = 2pq which suggests that μ = np as before and that σ 2 = npq, so that σ = % npq

(F) Examples ex 1  A fair die is tossed 12 times and the random variable, X, is the number of 6's that occur. Find the mean and standard deviation of the binomial distribution of X. (ANS:2,1.291) ex 2  Five percent of a batch of batteries are defective. A random sample of 80 batteries, with replacement, is taken. Find the mean and standard deviation of the number of defectives in the sample. (ANS:4,1.949)

(G) Classwork/Homework HL Math textbook, Chap30G, p745, Q1-5 SL Math textbook, Chap29F, p727, Q1-6