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More on Binomial Probability (Mean, SD. Histograms, Shape, etc.)

 

Example 1:

 

(a)  Construct a binomial probability distribution with the given parameters:   , 

 

(b)  Compute the mean and standard deviation of the distribution, using the methods of Sect. 6.1

 

(c)  Compute the mean and standard deviation, using the methods of this section.

 

(d)  Draw the probability histograms, comment on its shape, and label the mean on the histogram.

 

(a.) Construct a probability distribution for a binomial probability model with n = 9 and p = .75. Press STAT, select 1:Edit and clear L1 and L2.  Enter the values 0 through 9 into L1.  Press 2nd [QUIT].  To calculate the probabilities for each X-value in L1, first change the display mode so that the probabilities displayed will be rounded to 3 decimal places.  Press MODE and change from FLOAT to 3 and press 2nd  [QUIT]. 

 

Next press 2nd [DISTR] and select 0:binompdf( and type in 9  ,  .75 ) and press ENTER.  Store these probabilities in L2 by pressing STO 2nd  [L2].   

 

(b.)  Press STAT and highlight CALC.  Select 1:1-Var Stats, press ENTER and press 2nd [L1], 2nd [L2] ENTER to see the descriptive statistics.  

 

(c.)  Use the formulas for the mean and standard deviation of a binomial random variable.  The mean is   the standard deviation is:   

 

(d.)  To graph the binomial distribution, press 2nd [STAT PLOT] and press ENTER.  Turn ON Plot 1, select Histogram for Type, type in 2nd [L1] for Xlist and 2nd [L2] for Freq.  Adjust the graph window by pressing WINDOW and setting Xmin = 0, Xmax = 10, Xscl =1, Ymin = 0 and Ymax = .31.  Choosing ‘Xmax=10’ leaves some space at the right of the graph in order to complete the histogram.  The Ymax value was selected by looking through the values in L2 and then rounding the largest value UP to a convenient number.  Press GRAPH to view the histogram.

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Example 2:  According to the United States National Center for Health Statistics, there is a 98% probability that a 20-year-old male will survive to age 30. 


(a)  Using statistical software, such as Minitab, simulate taking 100 random samples of size 30 from this population.

 

(b) Using the results of the simulation, compute the probability that exactly 29 of the 30 males survive to age 30.

 

(c) Compute the probability that exactly 29 of the 30 males survive to age 30, using the binomial probability distribution.  Compare the results with part (b).    0.3340

 

(d) Using the results of the simulation, compute the probability that at most 27 or the 30 males survive to age 30.

 

(e) Compute the probability that at most 27 or the 30 males survive to age 30, using the binomial probability distribution.  Compare the results with part (d).   0.0217

 

(f)  Compute the mean number of male survivors in the 100 simulation of the probability experiment. Is it close to the expected value?

 

(g) Compute the standard deviation of the number of male survivors in the 100 simulations of the probability experiment.  Compare the result to the theoretical standard deviation of the probability distribution.

 

(h)  Did the simulation yield any unusual results?

 

 

(a.)  To generate random samples for this binomial model, press MATH, select PRB and select 7:randBin(.  This command requires three values: n, which is the sample size; p, the probability; and x, the number of samples.  For this example type in 30 , .98, 100).  Press ENTER.  It will take the calculator a few minutes to complete this stimulation.   

 

 

Store these probabilities in L1 by pressing STO 2nd [L1].     

 

(b.)  To use the results of the simulation to compute the probability that exactly 29 of the 30 males survive to age 30, construct a histogram of the simulation.  Press  2nd [STAT PLOT] and press ENTER.  Turn ON Plot 1, select Histogram for Type, type in 2nd [L1] for Xlist and 1 for Freq.  Adjust the graph window by pressing WINDOW and setting Xmin = 25, Xmax = 31, Xscl = 1, Ymin = 0 and Ymax = 60.  Press GRAPH to view the histogram.  Press TRACE and scroll through the bars until you reach the bar for ‘29’.  Take the frequency for that bar and divide it by 100 (the total number of simulations).  Your result is the probability that exactly 29 males in a sample of 30 males will survive to age 30.

 

 

In the simulation, the probability is 31 out of 100 or 31%.

 

(c.)  Press 2nd DISTR and select 0:binompdf and enter 30 , .98 , 29).

 

(d.)  Press GRAPH and the histogram of the simulation will appear.  Press TRACE and scroll through the bars for ‘28’, ‘29’, and ‘30’.  Sum the frequencies for these bars.  Divide this sum by 100.  This value is    The complement of this is   Subtract   from 1 to get

 

(e.)  Press 2nd DISTR and select A:binomcdf and enter 30 , .98 , 27).  This value is

 

(f-g).  First, calculate the mean and standard deviation of the 100 simulations.  Press STAT and highlight CALC.  Select 1:1-Var Stats, press ENTER and press 2nd [L1] ENTER to see the descriptive statistics.  Then, use the formulas for the mean and standard deviation of a binomial random variable.  The mean is:  ;  the standard deviation is: 

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Example 3: A probability distribution for the random variable X, the number of trials until the first success is observed, is called the geometric probability distribution.  It has the same requirements as the binomial distribution (page 339 in text book), except that the number of trials is not fixed.  Its probability distribution function (pdf) is        x = 1, 2, 3, …  where p is the probability of success.

 

(a)  What is the probability that Shaquille O’Neal takes three free throws before he makes one?  Over his career, he makes 53.6% of his free throws.  That is, find    0.1154

 

(b) Construct a probability distribution for the random variable X, the number of free throw attempts of Shaquille O’Neal before he makes a free throw.  Construct the distribution for x = 1, 2, 3, …, 10.  The probabilities are small for

 

(c) Compute the mean of the distribution, using the formula presented in Section 6.1.

 

(d) Compare the mean obtained in part (c) with the value .  Conclude that the mean of a geometric probability distribution is   .  How many free throws do we expect Shaq to take before we observe a made free throw?

 

 

(a.)  Suppose the probability that Shaquille O’Neal makes a free throw is .536.  To find the probability that the first free throw he makes occurs on his third shot, press 2nd DISTR and select D:geometpdf( and type in .536 , 3 ).

 

 

(b.)  Construct a probability distribution for a geometric probability model with p = .563.  Press STAT, select 1:EDIT and clear L1 and L2.  Enter the values 1 through 10 into L1.  Press 2nd [QUIT].  To calcutate the probabilities for each X-value in L1, first change the display mode so that the probabilities displayed will be rounded to 3 decimal places.  Press MODE and change from FLOAT to 3 and press 2nd [QUIT].

 

Next press 2nd DISTR and select D:geometpdf( and type in .563 , L1 ) and press ENTER.  Store these probabilities in L2 by pressing STO 2nd [L2}.

 

(c.)  Press STAT and highlight CALC. Select 1:1- Var Stats, press ENTER and press 2nd [L1] , 2nd [L2] ENTER to see the descriptive statistics.

 

(d.)  Calculate the mean of a geometric probability model using the formula: 

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