Chapter 17 Probability Models Binomial Probability Models Poisson Probability Models

Binomial Random Variables Binomial Probability Distributions

Binomial Random Variables Through 2/25/2014 NC State’s free-throw percentage is 65.1% (315 th out 351 in Div. 1). If in the 2/26/2014 game with UNC, NCSU shoots 11 free-throws, what is the probability that:  NCSU makes exactly 8 free-throws?  NCSU makes at most 8 free throws?  NCSU makes at least 8 free-throws?

“2-outcome” situations are very common Heads/tails Democrat/Republican Male/Female Win/Loss Success/Failure Defective/Nondefective

Probability Model for this Common Situation Common characteristics ◦ repeated “trials” ◦ 2 outcomes on each trial Leads to Binomial Experiment

Binomial Experiments n identical trials ◦ n specified in advance 2 outcomes on each trial ◦ usually referred to as “success” and “failure” p “success” probability; q=1-p “failure” probability; remain constant from trial to trial trials are independent

Binomial Random Variable The binomial random variable X is the number of “successes” in the n trials Notation: X has a B(n, p) distribution, where n is the number of trials and p is the success probability on each trial.

Binomial Probability Distribution

P(x) = p x q n-x n !n ! ( n – x )! x ! Number of outcomes with exactly x successes among n trials Rationale for the Binomial Probability Formula

P(x) = p x q n-x n !n ! ( n – x )! x ! Number of outcomes with exactly x successes among n trials Probability of x successes among n trials for any one particular order Binomial Probability Formula

Graph of p(x); x binomial n=10 p=.5; p(0)+p(1)+ … +p(10)=1 Think of p(x) as the area of rectangle above x p(5)=.246 is the area of the rectangle above 5 The sum of all the areas is 1

Binomial Distribution Example: Tennis First Serves A tennis player makes 60% of her first serves and attempts 4 first-serves. Assume the outcomes of first-serves are independent. What is the probability that exactly 3 of her first serves are successful?

15 Binomial Distribution Example Shanille O’Keal is a WNBA player who makes 25% of her 3- point attempts. Assume the outcomes of 3-point shots are independent. 1. If Shanille attempts 7 3-point shots in a game, what is the expected number of successful 3-point attempts? 2. Shanille’s cousin Shaquille O’Neal makes 10% of his 3-point attempts. If they each take 12 3-point shots, who has the smaller probability of making 4 or fewer 3-point shots? Shanille has the smaller probability.

Using binomial tables; n=20, p=.3 P(x  5) =.4164 P(x > 8) = 1- P(x  8)= =.1133 P(x < 9) = ? P(x  10) = ? P(3  x  7)=P(x  7) - P(x  2) = , 10, 11, …, 20 8, 7, 6, …, 0 =P(x  8) 1- P(x  9) =

Binomial n = 20, p =.3 (cont.) P(2 < x  9) = P(x  9) - P(x  2) = =.9165 P(x = 8) = P(x  8) - P(x  7) = =.1144

Color blindness The frequency of color blindness (dyschromatopsia) in the Caucasian American male population is estimated to be about 8%. We take a random sample of size 25 from this population. We can model this situation with a B(n = 25, p = 0.08) distribution. What is the probability that five individuals or fewer in the sample are color blind? Use Excel’s “=BINOMDIST(number_s,trials,probability_s,cumulative)” P(x ≤ 5) = BINOMDIST(5, 25,.08, 1) = What is the probability that more than five will be color blind? P(x > 5) = 1  P(x ≤ 5) =1  = What is the probability that exactly five will be color blind? P(x = 5) = BINOMDIST(5, 25,.08, 0) =

Probability distribution and histogram for the number of color blind individuals among 25 Caucasian males. B(n = 25, p = 0.08)

What are the mean and standard deviation of the count of color blind individuals in the SRS of 25 Caucasian American males? µ = np = 25*0.08 = 2 σ = √np(1  p) = √(25*0.08*0.92) = 1.36 p =.08 n = 10 p =.08 n = 75 µ = 10*0.08 = 0.8 µ = 75*0.08 = 6 σ = √(10*0.08*0.92) = 0.86 σ = √(75*0.08*0.92) = 2.35 What if we take an SRS of size 10? Of size 75?

Recall Free-throw question Through 2/25/14 NC State’s free-throw percentage was 65.1% (315 th in Div. 1). If in the 2/26/14 game with UNC, NCSU shoots 11 free- throws, what is the probability that: 1.NCSU makes exactly 8 free-throws? 2.NCSU makes at most 8 free throws? 3.NCSU makes at least 8 free-throws? 1. n=11; X=# of made free-throws; p=.651 p(8)= 11 C 8 (.651) 8 (.349) 3 = P(x ≤ 8)= P(x ≥ 8)=1-P(x ≤7) = =.4283

22 Poisson Probability Models The Poisson experiment typically models situations where rare events occur over a fixed amount of time or within a specified region Examples ◦ The number of cellphone calls per minute arriving at a cellphone tower. ◦ The number of customers per hour using an ATM ◦ The number of concussions per game experienced by the participants.

24 ◦ Properties of the Poisson experiment 1)The number of successes (events) that occur in a certain time interval is independent of the number of successes that occur in another time interval. 2)The probability of a success in a certain time interval is  the same for all time intervals of the same size,  proportional to the length of the interval. 3)The probability that two or more successes will occur in an interval approaches zero as the interval becomes smaller. Poisson Experiment

25 The Poisson Random Variable ◦ The Poisson random variable X is the number of successes that occur during a given time interval or in a specific region Probability Distribution of the Poisson Random Variable.

Poisson Prob Dist =1

Poisson Prob Dist =5

28 Example ◦ Cars arrive at a tollbooth at a rate of 360 cars per hour. ◦ What is the probability that only two cars will arrive during a specified one-minute period?  The probability distribution of arriving cars for any one- minute period is Poisson with = 360/60 = 6 cars per minute. Let X denote the number of arrivals during a one-minute period.

29 ◦ Example (cont.) ◦ What is the probability that at least four cars will arrive during a one-minute period? ◦ P(X>=4) = 1 - P(X<=3) = =.849