Chapter 17 Probability Models

math2200

I don’t care about my [free throw 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.

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)=?

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.

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

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-1 p– E (X) = 1/p– Var (X) = q/p2

• 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)

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.

The Binomial Model (cont.)

• There are

ways to have k successes in n trials. – Read nCk as “n choose k.”

!

! !n k

nC

k n k

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

np npq

!( )

!( )!x n xn n n

P X x p q wherex x x n x

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 .

• 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

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

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

!

xeP X x

x

E X SD X

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.

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.

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!

# of hits 0 1 2 3 4 5

# of cells with # of hits above 229 211 93 35 7 1

Poisson Fit 226.7 211.4 98.5 30.6 7.1 1.6

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.

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.

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)