Chapter 17 Probability Models

1. Chapter 17 Probability Models math2200

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

5. Bernoulli trials • Only two possible outcomes • Success or failure • Probability of success, denoted by p, is the same for every trial • The trials are independent • Examples • tossing a coin • Free throw in a basketball game

6. Independence • Be careful when sampling without replacement in finite population • Precisely, these draws are not independent • But if the size of the population is large enough, we can treat them as independent • Rule of thumb: the sample size is smaller than 10% of the population

7. 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) = • E(X) = • Var(X) =

8. 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) = • Var(X) =

9. 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

10. What is the probability that Shaq makes at least one successful throw in the first four attempts?

11. What is the probability that Shaq makes at least one successful throw in the first four attempts? • 1-P(NNNN) = 1-(1-0.451)4 = 0.9092 • P(X=1)+P(X=2)+P(X=3)+P(X=4)

12. 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. • Binom(n,p) • Let X be the number of success in n Bernoulli trials • p = probability of success • q = 1-p = probability of failure

13. The Binomial Model (cont.) • In n trials, there are ways to have k successes. • Read nCk as “n choose k.” • Note: n! = n x (n-1) x … x 2 x 1, and n! is read as “n factorial.”

14. The Binomial Model (cont.) Binomial probability model for Bernoulli trials: Binom(n,p) n = number of trials p = probability of success q = 1 – p = probability of failure X = number of successes in n trials

15. How do we find E(X) and Var(X)? • Use 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 outcome of each Bernoulli trial • E(Y1)=…=E(Yn)=p*1+q*0=p • Var(Y1)=…=Var(Yn)=(1-p)2 *p+(0-p)2 *q = pq

16. Mean and variance of sum • Suppose Y1,…,Yn are independent and have the same mean µ and variance σ2 • Let X = Y1+…+Yn • E(X) = E(Y1)+…+E(Yn)=nµ • Var(X) = Var(Y1)+…+Var(Yn)=nσ2

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

18. Normal approximation to Binomial • If X ~ Binomial(n,p), n=10000 • P(X<2000)=? • When dealing with a large number of trials in a Binomial situation, making direct calculations of the probabilities becomes tedious (or outright impossible). • When n is large, np is not too small or too big, then Binomial(n,p) looks similar to Normal with mean = np and variance = npq • P(X<2000)=P(Z<(2000-np)/sqrt(npq)) • Success/failure condition : np>=10 and nq>=10

19. Continuous Random Variables • When we use the Normal model to approximate the Binomial model, we are using a continuous random variable to approximate a discrete random variable. • So, when we use the Normal model, we no longer calculate the probability that the random variable equals a particular value, but only that it lies between two values.

20. Poisson model • For small p and large n, even when np<10, we can approximate Binomial(n,p) by Poisson(np) • Let λ=np, we can use Poisson model to approximate the probability. • Poisson(λ) • λ : mean number of occurrences • X: number of occurrences

21. The Poisson Model (cont.) • Although it was originally an approximation to the Binomial, the Poisson model is also used directly to model the probability of the occurrence of events for a variety of phenomena. • It’s a good model to consider whenever your data consist of counts of occurrences. • It requires only that the events be independent and that the mean number of occurrences stays constant.

22. More about Poisson model • 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 • Even though the events appear to cluster, the probability of another event occurring is still the same

23. 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. • Want to know if the Germans were targeting these districts or if the distribution was due to chance. • Clarke began by dividing an area into hundreds of tiny, equally sized plots.

25. Flying bomb hits on London • The average number of hits per square is then 537/576=.9323 hits per square • No need to move people from one sector to another, even after several hits!

26. Flying bomb hits on London • The average number of hits per square is then 537/576=.9323 hits per square • No need to move people from one sector to another, even after several hits!

27. 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.

28. What have we learned? • Geometric model • When we’re interested in the number of Bernoulli trials until the next success. • Binomial model • When we’re interested in the number of successes in a certain 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.

29. TI-83 • 2nd + VARS (DISTR) • pdf: P(X=x) • geometpdf(prob,x) • binompdf(n,prob,x) • poissonpdf(mean,x) • cdf: P(X<=x) • beometcdf(prob,x) • binomcdf(n,prob,x) • poissoncdf(mean,x)