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STAB22 Statistics I Lecture 18 1 Bernoulli Trial Bernoulli Trial: trial with only 2 outcomes E.g. True/False, Yes/No, Heads/Tails P(Success) = p, P(failure) = 1−p Value 1 0 Prob p 1‒p E.g. Fair coin flip, let Success = Heads Usually labeled Success (1) and Failure (0) P(1) = ½, & P(0) = 1− ½ = ½ Bernoulli trials form basis of many common probability models 2 Binomial Model Several Bernoulli trials, but only interested in total number of successes Example: # students who vote Yes for a proposal Binomial Setting: 1. Fixed number (n) of Bernoulli trials 2. Same probability of success (p ) for each trial 3. Bernoulli trials are independent Binomial Random Variable: X = # of successes in a Binomial setting 3 Binomial Distribution If X follows Binomial distribution with n trials and probability of success p X takes values from 0 to n (i.e. 0,1,…,n) Probabilities given by formula P( X x) n Cx p x (1 p) n x , for x 0,1,..., n n! where: n Cx , n ! n (n 1) x ! n x ! 2 1 Or, simply use software / probability tables (StatCrunch: Stat > Calculators > Binomial) 4 Example Multiple choice test has 10 questions, each with 4 choices: A,B,C or D. Student has not studied at all, but thinks he will give it a shot (i.e. answer at random). What is the probability model of his score? Does it fit the Binomial? Number of trials? Are they independent? Probability of success? Is the score a binomial RV? 5 Example (cont’d) Find prob. student’s score is 5/10 Find prob. student passes ( ≥5/10) Find prob. student gets at least 2 questions right ( ≥2/10) x P(x) 0 0.0563 1 0.1877 2 0.2816 3 0.2503 4 0.1460 5 0.0584 6 0.0162 7 0.0031 8 0.0004 9 0.0000 10 0.0000 6 Example This Halloween, you have a bag with 14 Snickers and 22 Mars bars. Neighbor's daughter comes trick-ortreating, and picks 3 bars at random. What is the probability model of # Mars bars? Does it fit the Binomial? Number of trials? Are they independent? Probability of success? Is the score a Binomial RV? Mean & Variance of Binomial Luckily, also have formulas for mean and variance of Binomial random variable If X is a Binomial RV, then: E X n p & V X n p 1 p 2 Mean increases with # of trials n and probability of success p Variance increases with # of trials n Variance is biggest when p = 1/2 Variance becomes small when p is close to 0 or 1 (Is that reasonable?) 8 Binomial Model for Different p p = 0.05 n = 10 p = 0.10 n = 10 p = 0.20 n = 10 p = 0.50 n = 10 p = 0.70 n = 10 9 Binomial Model for Different n p = 0.10 10 Example For multiple choice test (n = 10, p = 1/4) Find student’s expected test score Find student’s test score variance and standard deviation 11 Normal Approximation to Binomial Random variable X follows a Binomial with n=80 trials, and p=0.5 probability of success. You want to know the probability that X takes values from 41 to 60, i.e. P( 41 ≤ X ≤ 60 ). You have to calculate 20 probabilities using the Binomial probability formula, and add them up! P(X=41) + P(X=42) + … + P(x=60) Thankfully there is an easier way, using the Normal approximation to the Binomial! Normal Approximation to Binomial n p n p (1 p) 0.10 Approximate Binomial with a Normal with the same mean & standard deviation: 0.08 0.06 Use approximation when np>10 AND n(1−p)>10 (Success / Failure condition) 0.04 Normal ( μ=30, σ=√15=3.87) 0.02 The Normal distribution yields a good approximation of the Binomial for large n values 0.00 20 23 26 29 32 35 38 Binomial (n=60, p=0.5) Example The management of the Santoni Pizza found that 70 percent of its new customers return for another meal. For a week in which 80 new (first-time) customers dined at Santoni’s, what is the probability that 60 or more will return for another meal? Random Variable X = number of returning customers Binomial distribution: X is number of successes in 80 independent trials Binomial parameters: n=80, π=0.7 Want to find P( X ≥ 60 ) Example (cont’d) Using Binomial probability table (n=80, p=.7) … P(X ≥ 60) = 0.063 + 0.048 + 0.034 + … = 0.197 Example (cont’d) Using Normal Approximation 16