The Binomial Distribution Discrete Random Variables 1 Some
The Binomial Distribution Discrete Random Variables 1
Some Special Discrete Distributions � Sometimes we can use specific distributions to model situations � Some ◦ ◦ ◦ important discrete distributions are: Binomial Poisson Geometric Negative Binomial Hypergeometric 2
Binomial Distribution � One of several specific discrete probability distributions. � Handles probability problems with two outcomes (or problems that can be reduced to two outcomes). � Result of multiple Bernoulli trials.
Binomial Experiment Conditions � To have a binomial experiment, all four of the following properties must be true. 1. There a fixed number of trials n. 2. Each observation fall into one of just two categories (called success and failure). 3. The probability of a success is the same for each trial and is labeled, p. 4. The n trials are all independent. Note: Sometimes 3&4 are put together
Binomial Distribution � ◦ ◦ � A Binomial random variable X, denoted B(n, p), represents the count of successes in a binomial experiment where: n is the total number of trials. The parameter p is the probability of success on each trial. The count of successes X can be any whole number between 0 and n.
Example �A coin is flipped 5 times. Does this experiment fit the binomial setting? � Fixed number of trials? Yes, n = 5 � Independence? � Two Yes outcomes? Yes (heads & tails) � Probability of success remains constant? Yes
Finding Binomial Probabilities � Suppose we wanted to know the probability of getting two heads in 5 tosses? ◦ B(5, 0. 5) w/ P(X=2) � The sample space consists of 2 n possible outcomes. 25 = 32 possible outcomes S = { SSSSS, SSSSF, SSSFS, SSFSS, SFSSS, FSSSS, SSSFF, SSFSF, SSFFS, SFSSF, SFSFS, SFFSS, FSSSF, FSSFS, FSFSS, FFSSS, SSFFF, SFSFF, SFFSF, SFFFS, FSSFF, FSFSF, FSFFS, FFSSF, FFSFS, FFFSS, SFFFF, FSFFF, FFSFF, FFFSF, FFFFS, FFFFF }
Building the Binomial Formula � The first part of the binomial function gives us the number of ways of choosing exactly k successes among n trials. ◦ This is also called the number of possible combinations. ◦ This piece of the function is also known as the binomial coefficient. n “choose” k Where: k = 0, 1, 2, . . . , or n. k! = k(k – 1)(k – 2)(k – 3)…(2)(1)
Example Flip a coin five times. First, find out how many ways can we obtain 2 heads in the 5 flips where a head is a success. Let’s use the first part of the function
Example: Five coin flips � Thus, there are 10 ways to get exactly 2 heads in five flips. � Consider “heads” a success. Check this using the sample space… S = { SSSSS, SSSSF, SSSFS, SSFSS, SFSSS, FSSSS, SSSFF, SSFSF, SSFFS, SFSSF, SFSFS, SFFSS, FSSSF, FSSFS, FSFSS, FFSSS, SSFFF, SFSFF, SFFSF, SFFFS, FSSFF, FSFSF, FSFFS, FFSSF, FFSFS, FFFSS, SFFFF, FSFFF, FFSFF, FFFSF, FFFFS, FFFFF }
Binomial Probability Function � We know: ◦ There are 10 ways this can happen ◦ Probability of getting heads on a coin flip is 0. 5 ◦ Getting a head on the first flip has no affect on the second � Consider the sequences: ◦ SSFFF in five coin flips. ◦ SFFSF in five coin flips � Are these probabilities the same? ◦ Yes � We want to calculate the probability of one of these sequences happening.
Example: Five Coin Flips � Consider using the Multiplication Rule for independent events P(SSFFF) = P(S) * P(F) *P(F) P(SSFFF) = P(S)2 * P(F)3 P(SSFFF) = 0. 52 * 0. 53 P(SSFFF) = 0. 03125
Second Piece of the Function � To obtain the probability of any one specific outcome we use… P(“outcome”) = p#S (1 – p)#F Where p = probability of success 1 – p = probability of failure #S = number of successes (k) #F = number of failures (n – k)
Binomial Probability Function Combine the two pieces to obtain the function n = # of trials k = # of successes p = probability of success
Example: Five Coin Flips �
Binomial Probability Function � The first part of the function gives you the number of ways to arrange k successes in n trials. � The second part of the function calculates the probability of one of the possible outcomes � Note, the function calculates the probabilities of exactly k successes in n trails [P(X = k)]. � If you want to calculate cumulative probabilities you must: ◦ Use it multiple times. ◦ Use Technology
Binomial Mean and Variance � If we have a binomial distribution with n trials and the probability of success p, then
Ways to solve Binomial problems � Pencil and Paper : | � Tables : ( � Graphing � Excel in Minitab : ) : ) 18
Binomial example � � Samples of water have a 10% chance of containing a pollutant. Assume that the samples are independent with regard to the presence of the pollutant. Find the probability that, in the next 18 samples, exactly 2 contain the pollutant. Let X denote the number of samples that contain the pollutant in the next 18 samples analyzed. Then X is a binomial random variable with p = 0. 1 and n = 18. 19
Binomial example cont. � Determine the probability that at least 4 samples contain the pollutant. B(18, 0. 1). � Lets take a look at graphing this in Minitab 20
Binomial Minitab Example � � Graph of P(X≥ 4) for B(18, 0. 1) Steps: ◦ Go to graph menu → Probability Distribution Plot ◦ Select Single →Choose Binomial, enter parameters ◦ To select desired shaded area double click on bars of graph ◦ Click on the shaded area tab and enter what you would like to shade 21
Binomial example cont. determine the probability that 3 ≤ X ≤ 7. B(18, 0. 1) � Now 22
Binomial Table Example � The probability that on entering college, a student will graduate in 4 years is 0. 77. An academic advisor is advising 12 freshmen. � What is the probability of the following events? a) b) c) d) Exactly 10 of the 12 graduate? Less than half graduate? 8 or more graduate? Between 7 and 9 inclusive graduate?
Binomial Conditions First check if this situation fits the binomial setting. Ask: Fixed number of trials, n? Two outcomes? Do we have a probability of success, p? Are these n trials independent?
Building the Probability Distribution �I recommend building the table of all probabilities to help �I would recommend Excel for this � Start � Then by listing all possible values of X. list corresponding PMF (0) and CDF (1) values.
Binomial Probability Table � The completed Binomial Distribution table looks like this: Grad. P(X = x) P(X <= x) 0 1 2 3 4 5 6 7 8 9 10 11 12 0. 0000 0. 0002 0. 0014 0. 0073 0. 0285 0. 0818 0. 1712 0. 2547 0. 2558 0. 1557 0. 0434 0. 0000 0. 0002 0. 0016 0. 0089 0. 0374 0. 1192 0. 2904 0. 5450 0. 8009 0. 9566 1. 0000 26
Solution to (a) � Exactly 10 of the 12 graduate? � P(X = 10) � Using the table: Grad. P(X = x) P(X <= x) 0 1 2 3 4 5 6 7 8 9 10 11 12 0. 0000 0. 0002 0. 0014 0. 0073 0. 0285 0. 0818 0. 1712 0. 2547 0. 2558 0. 1557 0. 0434 0. 0000 0. 0002 0. 0016 0. 0089 0. 0374 0. 1192 0. 2904 0. 5450 0. 8009 0. 9566 1. 0000
Solution to (b) � What is the probability that less than half the students graduate. � P(X < 6) = P(X < 5) � Using the Table Grad. P(X = x) P(X <= x) 0 1 2 3 4 5 6 7 8 9 10 11 12 0. 0000 0. 0002 0. 0014 0. 0073 0. 0285 0. 0818 0. 1712 0. 2547 0. 2558 0. 1557 0. 0434 0. 0000 0. 0002 0. 0016 0. 0089 0. 0374 0. 1192 0. 2904 0. 5450 0. 8009 0. 9566 1. 0000
Solution to (c) � What is the probability that 8 or more of the students graduate? � P(X � > 8) = 1 – P(X < 7) Using the table ◦ 1 – 0. 1192 = 0. 8808 Grad. P(X = x) P(X <= x) 0 1 2 3 4 5 6 7 8 9 10 11 12 0. 0000 0. 0002 0. 0014 0. 0073 0. 0285 0. 0818 0. 1712 0. 2547 0. 2558 0. 1557 0. 0434 0. 0000 0. 0002 0. 0016 0. 0089 0. 0374 0. 1192 0. 2904 0. 5450 0. 8009 0. 9566 1. 0000
Solution to (d) � Between � P(7 7 and 9 (inclusive) graduate? < X < 9) � Using the table ◦ 0. 0818+0. 1712+0. 2547= 0. 5077 Grad. P(X = x) P(X <= x) 0 1 2 3 4 5 6 7 8 9 10 11 12 0. 0000 0. 0002 0. 0014 0. 0073 0. 0285 0. 0818 0. 1712 0. 2547 0. 2558 0. 1557 0. 0434 0. 0000 0. 0002 0. 0016 0. 0089 0. 0374 0. 1192 0. 2904 0. 5450 0. 8009 0. 9566 1. 0000
Graduation Example � The probability that on entering college, a student will graduate in 4 years is 0. 77. An academic advisor is advising 12 freshmen. � Find the Mean and SD of this variable
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