Chapter 17 Probability Models

1. Chapter 17 Probability Models math2200

2. I don’t care about my [freethrow shooting] percentages. I keep telling everyone that I make them when they count.-- Shaquille O'Neal, in post-game interviews recorded by WOAI-TV on November 7, 2003.

3. O’Neal’s free throws • Suppose Shaq shoots 45.1% on average . Let X be the number of free throws Shaq needs to shoot until he makes one.Pr (X=2)=? Pr (X=5)=? E(X)=?

4. Bernoulli trials • Only two possible outcomes • Success or failure • Probability of success, denoted by p the same for every trial • The trials are independent • Examples • tossing a coin • Free throw in a basketball game assuming every time the player starts all over.

5. Can we model drawing without replacement by Bernoulli trials? • The draws are not independent when sampling without replacement in finite population. But they are be treated as independent if the population is large • Rule of thumb: the sample size is smaller than 10% of the population

6. Geometric model • How long does it take to achieve a success in Bernoulli trials? • A Geometric probability model tells us the probability for a random variable that counts the number of Bernoulli trials until the first success • Geom(p) • p = probability of success • q = 1-p = probability of failure • X : number of trials until the first success occurs • P (X=x) = qx-1p • E (X) = 1/p • Var (X) = q/p2

7. What is the probability that Shaq makes his first free throw in the first four attempts? • 1-P(NNNN) = 1-(1-0.451)4 = 0.9092or P(X=1)+P(X=2)+P(X=3)+P(X=4)

8. Binomial model • A Binomial model tells us the probability for a random variable that counts the number of successes in a fixed number of Bernoulli trials.

9. The Binomial Model (cont.) • There are ways to have k successes in n trials. • Read nCk as “n choose k.”

10. The Binomial Model (cont.) • Binom (n, p) n = number of trials p = probability of success q = 1 – p = probability of failure X = number of successes in n trials

11. How do we find E(X) and Var(X)? • Find P(X=x) directly • Binomial random variable can be viewed as the sum of the outcome of n Bernoulli trials • Let Y1,…, Yn be the outcomes of n Bernoulli trials • E (Y1) =…= E (Yn) = p*1+q*0=p, E(X) = np • Var (Y1) =…= Var (Yn) = (1-p)2 *p+(0-p)2 *q = pq,Var (X) = npq. • In general if Y1,…,Yn are independent and have the same mean µ and variance σ2 and X = Y1+…+Yn, then E(X) = E(Y1)+…+E(Yn)=nµ and Var(X) = Var(Y1)+…+Var(Yn)=nσ2 .

12. If Shaq shoots 20 free throws, what is the probability that he makes no more than two? • Binom(n, p), p=0.451, n=20P(X=0 or 1 or 2) = P(X=0) + P(X=1) + P(X=2) = 0.0009

13. Normal approximation to Binomial • If X ~ Binomial (n, p), n =10000, p =0.451, P(X<2000)=? • When Success/failure condition (np >= 10 and nq>=10) is satisfied, Binomial (n,p) can be approximated by Normal with mean np and variance npq. • P (X<2000)=P ( Z< (2000-np) / sqrt (npq)) = P(Z< -50.4428) =normalcdf (-1E99, -50.4428 ,0,1) = 0

14. Poisson model • Binomial(n,p) is approximated by Poisson(np) if np<10. • Let λ=np, we can use Poisson model to approximate the probability. • Poisson(λ) • λ : mean number of occurrences • X: number of occurrences

15. Poisson Model (cont.) • The Poisson model is also used directly to model the probability of the occurrence of events. • It scales to the sample size • The average occurrence in a sample of size 35,000 is 3.85 • The average occurrence in a sample of size 3,500 is 0.385 • Occurrence of the past events doesn’t change the probability of future events.

16. An application of Poisson model • In 1946, the British statistician R.D. Clarke studied the distribution of hits of flying bombs in London during World War II. • Were targeted or due to chance.

17. Flying bomb (cont’) • The average number of hits per square is then 537/576=.9323 hits per square. Given the number of hits following a Poisson ModelP (X=0) = [e^(-0.923)*(-0.923)^0] / 0! = 0.3936470.393647* 576 = 226.7 • No need to move people from one sector to another, even after several hits!

18. What Can Go Wrong? • Be sure you have Bernoulli trials. • You need two outcomes per trial, a constant probability of success, and independence. • Remember that the 10% Condition provides a reasonable substitute for independence. • Don’t confuse Geometric and Binomial models. • Don’t use the Normal approximation with small n. • You need at least 10 successes and 10 failures to use the Normal approximation.

19. What have we learned? • Geometric model • When we’re interested in the number of Bernoulli trials until the first success. • Binomial model • When we’re interested in the number of successes in a fixed number of Bernoulli trials. • Normal model • To approximate a Binomial model when we expect at least 10 successes and 10 failures. • Poisson model • To approximate a Binomial model when the probability of success, p, is very small and the number of trials, n, is very large.

20. TI-83 • 2nd + VARS (DISTR) • pdf: P(X=x) when X is a discrete r.v. • geometpdf(prob,x) • binompdf(n,prob,x) • poissonpdf(mean,x) • cdf: P(X<=x) • geometcdf(prob,x) • binomcdf(n,prob,x) • poissoncdf(mean,x)