Probability Models Binomial, Geometric, and Poisson Probability
Models
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Binomial Random Variables Binomial Probability
Distributions
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Binomial Random Variables Through 2/10/2015 NC States
free-throw percentage is 67.4% (231 st out 351 in Div. 1). If in
the 2/11/2015 game with UVA, 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?
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2-outcome situations are very common Heads/tails
Democrat/Republican Male/Female Win/Loss Success/Failure
Defective/Nondefective
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Probability Model for this Common Situation Common
characteristics repeated trials 2 outcomes on each trial Leads to
Binomial Experiment
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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
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Classic binomial experiment: tossing a coin a pre-specified
number of times Toss a coin 10 times Result of each toss: head or
tail (designate one of the outcomes as a success, the other as a
failure; makes no difference) P(head) and P(tail) are the same on
each toss trials are independent if you obtained 9 heads in a row,
P(head) and P(tail) on toss 10 are same as P(head) and P(tail) on
any other toss (not due for a tail on toss 10)
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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.
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Binomial Probability Distribution
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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
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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
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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
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Binomial Distribution Example: Pepsi vs Coke In a taste test of
Pepsi vs Coke, suppose 25% of tasters can correctly identify which
cola they are drinking. If 12 tasters participate in a test by
drinking from 2 cups in which 1 cup contains Coke and the other cup
contains Pepsi, what is the probability that exactly 5 tasters will
correctly identify the colas?
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16 Binomial Distribution Example Shanille OKeal 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. Shanilles cousin Shaquille ONeal 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.
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 Excels
=BINOMDIST(number_s,trials,probability_s,cumulative) P(x 5) =
BINOMDIST(5, 25,.08, 1) = 0.9877 What is the probability that more
than five will be color blind? P(x > 5) = 1 P(x 5) =1 0.9877 =
0.0123 What is the probability that exactly five will be color
blind? P(x = 5) = BINOMDIST(5, 25,.08, 0) = 0.0329
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Probability distribution and histogram for the number of color
blind individuals among 25 Caucasian males. B(n = 25, p =
0.08)
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What are the mean and standard deviation of the count of color
blind individuals in the SRS of 25 Caucasian American males? = np =
25*0.08 = 2 = np(1 p) = (25*0.08*0.92) = 1.36 p =.08 n = 10 p =.08
n = 75 = 10*0.08 = 0.8 = 75*0.08 = 6 = (10*0.08*0.92) = 0.86 =
(75*0.08*0.92) = 2.35 What if we take an SRS of size 10? Of size
75?
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Recall Free-throw question Through 2/10/15 NC States free-throw
percentage was 67.4% (231 st in Div. 1). If in the 2/11/15 game
with UVA, 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=.674 p(8)= 11 C 8 (.674) 8 (.326) 3 =.243 2.
P(x 8)=.750 3. P(x 8)=1-P(x 7) =1-.5064 =.4936
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23 Geometric Random Variables Geometric Probability
Distributions Through 2/10/2015 NC States free-throw percentage is
67.4 (231 st of 351 in Div. 1). In the 2/11/2015 game with UVA what
is the probability that the first missed free- throw by the Pack
occurs on the 5 th attempt?
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24 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 The binomial rv
counts the number of successes in the n trials
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25 The Geometric Model A geometric random variable counts the
number of trials until the first success is observed. A geometric
random variable is completely specified by one parameter, p, the
probability of success, and is denoted Geom(p). Unlike a binomial
random variable, the number of trials is not fixed
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26 The Geometric Model (cont.) Geometric probability model for
Bernoulli trials: Geom(p) p = probability of success q = 1 p =
probability of failure X = # of trials until the first success
occurs p(x) = P(X = x) = q x-1 p, x = 1, 2, 3, 4,
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27 Example The American Red Cross says that about 11% of the
U.S. population has Type B blood. A blood drive is being held in
your area. 1. How many blood donors should the American Red Cross
expect to collect from until it gets the first donor with Type B
blood? Success=donor has Type B blood X=number of donors until get
first donor with Type B blood
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28 Example (cont.) The American Red Cross says that about 11%
of the U.S. population has Type B blood. A blood drive is being
held in your area. 2. What is the probability that the fourth blood
donor is the first donor with Type B blood?
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29 Example (cont.) The American Red Cross says that about 11%
of the U.S. population has Type B blood. A blood drive is being
held in your area. 3. What is the probability that the first Type B
blood donor is among the first four people in line?
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32 Example Shanille OKeal is a WNBA player who makes 25% of her
3-point attempts. 1. The expected number of attempts until she
makes her first 3-point shot is what value? 2. What is the
probability that the first 3-point shot she makes occurs on her 3
rd attempt?
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Question from first slide Through 2/10/2015 NC States
free-throw percentage was 67.4%. In the 2/11/2015 game with UVA
what is the probability that the first missed free-throw by the
Pack occurs on the 5 th attempt? Success = missed free throw
Success p = 1 -.674 =.326 p(5) =.674 4 .326 =.0673 33
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34 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.
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36 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
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37 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.
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Poisson Prob Dist =1
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Poisson Prob Dist =5
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40 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.
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41 Example (cont.) What is the probability that at least four
cars will arrive during a one-minute period? P(X>=4) = 1 -
P(X