Chapter Five The Binomial Probability Distribution and Related Topics Understandable Statistics Seventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Five The Binomial Probability Distribution and Related Topics

Statistical Experiment A statistical experiment or observation is any process by which an measurements are obtained

Examples of Statistical Experiments Counting the number of books in the College Library Counting the number of mistakes on a page of text Measuring the amount of rainfall in your state during the month of June

a quantitative variable that assumes a value determined by chance Random Variable a quantitative variable that assumes a value determined by chance

Discrete Random Variable A discrete random variable is a quantitative random variable that can take on only a finite number of values or a countable number of values. Example: the number of books in the College Library

Continuous Random Variable A continuous random variable is a quantitative random variable that can take on any of the countless number of values in a line interval. Example: the amount of rainfall in your state during the month of June

Probability Distribution an assignment of probabilities to the specific values of the random variable or to a range of values of the random variable

Probability Distribution of a Discrete Random Variable A probability is assigned to each value of the random variable. The sum of these probabilities must be 1.

Probability distribution for the rolling of an ordinary die x P(x) 1 2 3 4 5 6

Features of a Probability Distribution x P(x) 1 2 3 4 5 6 Probabilities must be between zero and one (inclusive)  P(x) =1

Probability Histogram P(x) | | | | | | | 1 2 3 4 5 6

Mean and standard deviation of a discrete probability distribution Mean =  = expectation or expected value, the long-run average Formula:  =  x P(x)

Standard Deviation

Finding the mean:  =  x P(x) = 1.4 0 .3 1 .3 .3 2 .2 .4 3 .1 4 .1 x P(x) x P(x) 0 .3 1 .3 2 .2 3 .1 4 .1 .3 .4  =  x P(x) = 1.4 1.4

Finding the standard deviation x P(x) x –  ( x – ) 2 ( x – ) 2 P(x) 0 .3 1 .3 2 .2 3 .1 4 .1 .588 .048 .072 .256 .676 – 1.4 – 0.4 .6 1.6 2.6 1.96 0.16 0.36 2.56 6.76 1.64

Standard Deviation 1.28

Linear Functions of a Random Variable If a and b are any constants and x is a random variable, then the new random variable L = a + bx is called a linear function of a random variable.

If x is a random variable with mean  and standard deviation , and L = a + bx then: Mean of L =  L = a + b  Variance of L = L 2 = b2  2 Standard deviation of L =  L= the square root of b2  2 = b 

If x is a random variable with mean = 12 and standard deviation = 3 and L = 2 + 5x Find the mean of L. Find the variance of L. Find the standard deviation of L. L = 2 + 5   Variance of L = b2  2 = 25(9) = 225 Standard deviation of L = square root of 225 = 

Independent Random Variables Two random variables x1 and x2 are independent if any event involving x1 by itself is independent of any event involving x2 by itself.

If x1 and x2 are a random variables with means and and variances and  If W = ax1 + bx2 then: Mean of W =  W = a + b  Variance of W = W 2 = a2  12 + b2  2 Standard deviation of W =  W= the square root of a2  12 + b2  2

Given x1, a random variable with 1 = 12 and  1 = 3 and x2 is a random variable with  2 = 8 and  2 = 2 and W = 2x1 + 5x2. Find the mean of W. Find the variance of W. Find the standard deviation of W. Mean of W = 2(12)+ 5(8) = 64 Variance of W = 4(9) + 25(4) = 136 Standard deviation of W= square root of 136  11.66

Binomial Probability

Features of a Binomial Experiment 1. There are a fixed number of trials. We denote this number by the letter n.

Features of a Binomial Experiment 2. The n trials are independent and repeated under identical conditions.

Features of a Binomial Experiment 3. Each trial has only two outcomes: success, denoted by S, and failure, denoted by F.

Features of a Binomial Experiment 4. For each individual trial, the probability of success is the same. We denote the probability of success by p and the probability of failure by q. Since each trial results in either success or failure, p + q = 1 and q = 1 – p.

Features of a Binomial Experiment 5. The central problem is to find the probability of r successes out of n trials.

Binomial Experiments Repeated, independent trials Number of trials = n Two outcomes per trial: success (S) and failure (F) Number of successes = r Probability of success = p Probability of failure = q = 1 – p

A sharpshooter takes eight shots at a target A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. Is this a binomial experiment?

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. success = failure =

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. success = hitting the target failure = not hitting the target

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. Probability of success = Probability of failure =

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. Probability of success = 0.70 Probability of failure = 1 – 0.70 = 0.30

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. In this experiment there are n = _____ trials.

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. In this experiment there are n = _8__ trials.

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. We wish to compute the probability of six successes out of eight trials. In this case r = _____.

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. We wish to compute the probability of six successes out of eight trials. In this case r = _ 6__.

Binomial Probability Formula

Calculating Binomial Probability Given n = 6, p = 0.1, find P(4):

Calculating Binomial Probability A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. n = 8, p = 0.7, find P(6):

Table for Binomial Probability Appendix II

Using the Binomial Probability Table Find the section labeled with your value of n. Find the entry in the column headed with your value of p and row labeled with the r value of interest.

Using the Binomial Probability Table n = 8, p = 0.7, find P(6):

Find the Binomial Probability Suppose that the probability that a certain treatment cures a patient is 0.30. Twelve randomly selected patients are given the treatment. Find the probability that: a. exactly 4 are cured. b. all twelve are cured. c. none are cured. d. at least six are cured.

Exactly four are cured: n = r = p = q =

Exactly four are cured: n = 12 r = 4 p = 0.3 q = 0.7 P(4) = 0.231

All are cured: n = 12 r = 12 p = 0.3 q = 0.7 P(12) = 0.000

None are cured: n = 12 r = 0 p = 0.3 q = 0.7 P(0) = 0.014

At least six are cured: r = ?

At least six are cured: r = 6, 7, 8, 9, 10, 11, or 12 P(6) = .079

At least six are cured: P( 6, 7, 8, 9, 10, 11, or 12) = .079 + .029 + .008 + .001 + .000 + .000 + .000 = .117

Graph of a Binomial Distribution Binomial distribution for n = 4, p = 0.4:

Graph of a Binomial Distribution Binomial distribution for n = 4, p = 0.4: P( r ) .4 .3 .2 .1 0 1 2 3 4 r

Mean and Standard Deviation of a Binomial Distribution

Mean and standard deviation of the binomial distribution Find the mean and standard deviation of the probability distribution for tossing four coins and observing the number of heads appearing. n = 4 p = 0.5 q = 1 – p = 0.5

Mean and standard deviation of the binomial distribution

Find the mean and standard deviation: A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. n = 8, p = 0.7

Geometric Distribution A probability distribution to determine the probability that success will occur on the nth trial of a binomial experiement

Geometric Distribution Repeated binomial trials Continue until first success Find probability that first success comes on nth trial Probability of success on each trial = p

Geometric Probability

A sharpshooter normally hits the target 70% of the time. Find the probability that her first hit is on the second shot. Find the mean and the standard deviation of this geometric distribution.

A sharpshooter normally hits the target 70% of the time. Find the probability that her first hit is on the second shot. P(2)=p(1-p) n-1 = .7(.3)2-1 = 0.21 Find the mean  = 1/p = 1/.7 1.43 Find the standard deviation

Poisson Distribution A probability distribution where the number of trials gets larger and larger while the probability of success gets smaller and smaller

Poisson Distribution Two outcomes : success and failure Outcomes must be independent Compute probability of r occurrences in a given time, space, volume or other interval  (Greek letter lambda) represents mean number of successes over time, space, area

Poisson Distribution

The mean number of people arriving per hour at a shopping center is 18. Find the probability that the number of customers arriving in an hour is 20. r = 20  = 18 Find P(20) e = 2.7183

The mean number of people arriving per hour at a shopping center is 18.

Poisson Probability Distribution Table Table 4 in Appendix II provides the probability of a specified value of r for selected values of .

Using the Poisson Table  = 18, find P(20):

Poisson Approximation to the Binomial Distribution The Poisson distribution can be used as a probability distribution for “rare” events.

“Rare” Event The number of trials (n) is large and the probability of success (p) is small.

If n  100 and np < 10, then The distribution of r (the number of successes) has a binomial distribution which is approximated by a Poisson distribution . The mean  = np.

Use the Poisson distribution to approximate the binomial distribution: Find the probability of at most 3 successes.

Using the Poisson to approximate the binomial distribution for n = 240 and p = 0.02 Note that n  100 and np = 4.8 < 10, so the Poisson distribution can be used to approximate the binomial distribution. Find the probability of at most 3 successes: Since  = np = 4.8, we use Table 4 to find P( r  3) =.0082 + .0395 + . 0948 + . 1517 = .2942