# M15- Binomial Distribution 1  Department of ISM, University of Alabama, 1992-2003 Lesson Objectives  Learn when to use the Binomial distribution.  Learn.

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M15- Binomial Distribution 1  Department of ISM, University of Alabama, 1992-2003 Lesson Objectives  Learn when to use the Binomial distribution.  Learn how to calculate probabilities for the Binomial using the formula and the two tables in the book, and in Excel. Bernoulli and Binomial Distributions Chapter 6.2, 7.4

M15- Binomial Distribution 2  Department of ISM, University of Alabama, 1992-2003  Two people in different rooms.  “A” is shown one of the five cards, selected randomly. “A” transmits his thoughts.  “B” selects the card she thinks is being sent to her, and records it.  If the two cards match, a success occurs. Does a person have ESP? Experiment: X= 1  success = 0  failure Bernoulli (  = ____ ) P(X = 1) = ___ =  P(X = 0) = ___ = 1-  X ~

M15- Binomial Distribution 3  Department of ISM, University of Alabama, 1992-2003 A discrete data distribution used to describe a population of binary variable values. Binary  one of only two outcomes can occur, coded as “0” or “1” Bernoulli Distribution

M15- Binomial Distribution 4  Department of ISM, University of Alabama, 1992-2003 Bernoulli distribution,  “one” trial. k 0 1 k 0 1 P(X=k)  =  = The mean of of the Bernoulli is The standard deviation is Bernoulli has one parameter:  = the probability of success. Bernoulli (  ) X ~ 1 ­  

M15- Binomial Distribution 5  Department of ISM, University of Alabama, 1992-2003 Bernoulli distribution,  “one” trial. k 0 1 k 0 1 P(X=k) 1 ­     2 = (0 -  ) 2 (1 -  ) + (1 -  ) 2  =  2 (1 -  ) + (1 -  ) 2  =  2 (1 -  ) + (1 -  ) 2  =  (1 -  ) [  + (1 -  ) ] =  (1 -  ) [  + (1 -  ) ] =  1 -  =  1 -   =  (1 -  ) 0(1-  ) + 1  =  = How are the “mean” and “standard deviation” determined? Once we know these results, we don’t need to derive it again. Use the same equations as the previous section. a little algebra...

M15- Binomial Distribution 6  Department of ISM, University of Alabama, 1992-2003 Examples of Bernoulli Variables Sex(male or female) Major(business or not business) Defective?(defective or non-defective) Response to a T-F question (true or false) Where student lives (on-campus or off-campus) Credit application result (accept or deny) Own home?(own or rent) Course result(Pass or Fail)

M15- Binomial Distribution 7  Department of ISM, University of Alabama, 1992-2003 Bar Chart of Population for ESP X 01  1–  0 1.0

M15- Binomial Distribution 8  Department of ISM, University of Alabama, 1992-2003 A discrete data distribution used to model a population of counts for “n” independent repetitions of a Bernoulli experiment. Binomial Distribution Conditions: 1.a fixed number of trials, “n”. 2.all n trials must be independent of each other. 3.the same probability of success on each trial. 4.X = count of the number of successes.

M15- Binomial Distribution 9  Department of ISM, University of Alabama, 1992-2003 Binomial distribution,  “n trials” k 0 1 2 3  n k 0 1 2 3  n P(X = k)  =  = The mean of of the Binomial is The standard deviation is Binomial has two parameters: n = the fixed number of trials,  = the probability of success for each trial. to be determined X ~ Bino( n,  )

M15- Binomial Distribution 10  Department of ISM, University of Alabama, 1992-2003 Computing Binomial probabilities  Formula gives P(X = k), the probability for exactly one value.  Tables Table A.1 gives P(X = k), the probability for exactly one value.  Excel(BINOMDIST function), gives either individual or cumulative. or Table A.2 gives P(X < k), the cumulative probability for X = 0 through X = k. For selected values of n and .

M15- Binomial Distribution 11  Department of ISM, University of Alabama, 1992-2003 Examples of Binomial Variables  A count of the number of females in a sample of 80 fans at a Rolling Stones concert.  A count of the number of defectives in sample of 50 tires coming off a production line.  A count of the number of the number of corrects answers on 10 true-false questions for which everybody guessed.  A count of the number of credit applications that are denied from a sample of 200.

M15- Binomial Distribution 12  Department of ISM, University of Alabama, 1992-2003 Are these situation Binomial?  A count of the number of defectives tires coming off a production line in one year.  Count of people choosing Dr Pepper over Pepsi in a blind taste-test with 20 people.  A count of the number of playing cards that are diamonds in a sample of 13 cards.  A count of the number of credit applications that are accepted from a sample of 200.  A count of the number of fans at a Stones concert needed until we find 50 females.

M15- Binomial Distribution 13  Department of ISM, University of Alabama, 1992-2003 If the population of X-values has a binomial data distribution, then the proportion of the population having the value k is given by: This is also the probability that a single value of X will be exactly equal to x. P ( X = k ) = ( ) nknk p k (1 – p) n–k for k = 0, 1, 2,..., n

M15- Binomial Distribution 14  Department of ISM, University of Alabama, 1992-2003 P ( X = k ) = ( ) nknk p k (1 – p) n–k Big “X” is the “random variable.” Little “k” is a “specific value” of Big X. Example: P( X = k)  P( Count = 4) Example: P( X = k)  P( Count = 4)

M15- Binomial Distribution 15  Department of ISM, University of Alabama, 1992-2003 P ( X = k ) = ( )( ) nknk p k (1 – p) n–k ( ) nk = n! = 0! =

M15- Binomial Distribution 16  Department of ISM, University of Alabama, 1992-2003 P ( X = x ) = ( ) nxnx p x (1 – p) n–x These are the possible values of X; each value has its own probability. x = 0, 1, 2,..., n

M15- Binomial Distribution 17  Department of ISM, University of Alabama, 1992-2003 X = a count of the number of successes. X ~ Bino(n=5,  =.20) Does a person have ESP? Experiment: Repeat the experiment 5 times. one Find the probability of exactly one success. P(X=1) = 5151.20.80 14  = 5.20.4096 =.4096 P(X=3) =

M15- Binomial Distribution 18  Department of ISM, University of Alabama, 1992-2003 Bino(n=5,  =.20) ESP Experiment two or fewer Find the probability of two or fewer successes. + 5.20.4096 P(X < 2) = P(X = 0) + P(X = 1) + P(X = 2) 5050.20.80 05  = + 5151.20.80 14  + 5252.20.80 23  = 1 1.32768+ 10.04.5120 =.32768 +.4096 +.2048.94208 =.94208

Table A.1 gives P( X = k ) P( X = 2 ) = Bino(n=5,  =.20) ESP Experiment.2048.9421 P( X < 2 ) = Table A.2 gives P( X < k ) Cumulative Individual k:  0.2.3277.4096.2048.0512.0064.0003 012345012345 n = 5 k:  0.2 012345012345 n = 5 1.0000.3277.7373.9421.9933.9997

M15- Binomial Distribution 20  Department of ISM, University of Alabama, 1992-2003 k 0.2.1678.5033.7969.9437.9896.9988.9999 1.000- 1.0000 012345678012345678 For the multiple choice test with 8 questions, 5 choices for each, find the probability of getting two or fewer correct. About 80% of the class should have two or fewer correct; about 20% should have three or more correct. About 80% of the class should have two or fewer correct; about 20% should have three or more correct. Table A.2 gives P( X £ k ) Bino(n=8,  =.20)

M15- Binomial Distribution 21  Department of ISM, University of Alabama, 1992-2003 Table A.2 gives P( X £ k ) Bino(n=8,  =.20) 012345678012345678.1678.5033.7969.9437.9896.9988.9999 1.000- 1.0000 k 0.2 For the multiple choice test, f ind the probability of more the one but no more than three correct. Same question as... “two or more but less than four;” or “exactly two or three.”

M15- Binomial Distribution 22  Department of ISM, University of Alabama, 1992-2003 k P( X  3) = 1.000 – P(X £ 2) = 1.000 –.7969 =.2031 Table A.2 gives P( X £ k ) Bino(n=8,  =.20) 012345678012345678.1678.5033.7969.9437.9896.9988.9999 1.000- 1.0000 k 0.2 For the multiple choice test find the probability of three or more. Same question as... “more than two”

M15- Binomial Distribution 23  Department of ISM, University of Alabama, 1992-2003 P( 3 or more correct ) = P( more than 3 correct ) = 012345678012345678.1678.5033.7969.9437.9896.9988.9999 1.000- 1.0000 k 0.2 Watch the wording! Table A.2 gives P( X £ k ) Bino(n=8,  =.20)

M15- Binomial Distribution 24  Department of ISM, University of Alabama, 1992-2003 Table A.2 gives P( X £ c ) Bino(n=8,  =.20) 012345678012345678.1678.5033.7969.9437.9896.9988.9999 1.000- 1.0000 k 0.2 Find the probability of at least one correct.

M15- Binomial Distribution 25  Department of ISM, University of Alabama, 1992-2003 X = a count of the number of successes. X ~ Bino(n=20,  =.20) Does a person have ESP? Experiment: Repeat the experiment 20 times. Beth and Mike matched 10 times? Is this unusual? P(X  10) = = = This is a rare event! The evidence indicates that they do better than guessing. The probability of observing a result this extreme is 0.0026.

M15- Binomial Distribution 26  Department of ISM, University of Alabama, 1992-2003 X = a count of the number of successes. X ~ Does a person have ESP? Bino(n=20,  =.20) Expected value  = _______ Standard deviation  =

M15- Binomial Distribution 27  Department of ISM, University of Alabama, 1992-2003 Free throws  Are shots independent?  Evidence says yes.  Suppose probability of Mo making one free throw is.70.  Mo will shoot 9 shots.

M15- Binomial Distribution 28  Department of ISM, University of Alabama, 1992-2003 Let X = count of shots made, Find probability he misses fewer than three. So, let Y = count of misses. Y ~ B ( ) n = 9,  =.70 n = 9,  = _____ X ~ B (  )

M15- Binomial Distribution 29  Department of ISM, University of Alabama, 1992-2003 Y 0 1 2 3 4 5 6 7 8 9 Misses X 9 8 7 6 5 4 3 2 1 0 Hits P( Y < 3 ) = ?.70 Count  P( Y £ 2 ) = P( X > 6 ) = 1 - P( X £ 6 ) == Lookup in Tables.

M15- Binomial Distribution 30  Department of ISM, University of Alabama, 1992-2003 Moral: Pay attention to what you are counting!

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