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CHAPTER 5 THE BINOMIAL PROBABILITY DISTRIBUTION
AND RELATED TOPICS
BINOMIAL PROBABILITY DISTRIBUTIONS (SECTIONS 5.2, 5.3 OF
UNDERSTANDABLE STATISTICS)
The binomial probability distribution is a discrete probability distribution controlled by the number of
trials, n, and the probability of success on a single trial, p.
MINITAB has three main commands for studying probability distributions.
The PDF (probability density function) gives the probability of a specified value for a discrete
distribution.
The CDF (cumulative distribution function) for a value X gives the probability a random variable with
distribution specified in a subcommand is less than or equal to X.
The INVCDF gives the inverse of the CDF. In other words, for a probability P, INVCDF returns the
value X such that P ≈ CDF(X). In this case of a binomial distribution. INVCDF often gives the two
values of X for which P lies between the respective CDF(X).
The three commands PDF, CDF, and INVCDF apply to many probability distributions. To apply them to a
binomial distribution, we need to use the menu selections.
CalchProbability distributionshBinomial
Dialog Box Responses
Select Probability for PDF; Cumulative probability for CDF;
Inverse cumulative probability for INVCDF
Number of trials: use the value of n in a binomial experiment,
Probability of success: use the value of p, the probability of success on a single trial,
Input column: Put the values of r, the number of successes in a binomial experiment in a
column such as C1. Select an optional storage column.
Note: MINITAB uses X instead of r to count the number of successes,
Input constant: Instead of putting values of r in a column, you can type a specific value of
r in the dialog box.
Example
A surgeon performs a difficult spinal column operation. The probability of success of the operation is
p = 0.73. Ten such operations are scheduled. Find the probability of success for 0 through 10 successes out of
these ten operations.
First enter the possible values of r, 0 through 10, in C1 and name the column r. We will put the
probabilities in C2, so name the column P(r).
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Fill in the dialog box as shown below.
Then use the hDatahDisplay data command.
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Next use the CDF command to find the probability of 5 or fewer successes. In this case use the option
for an input constant of 5. Leave Optional storage blank. The output will be P(r ≤ 5). Note that
MINITAB uses X in place of r.
The results follow.
Finally use INVCDF to determine how many operations should be performed in order for the probability of
that many or fewer successes to be 0.5. We select Inverse cumulative probability. Use .5 as the input constant.
The results follow.
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221
LAB ACTIVITIES FOR BINOMIAL PROBABILITY DISTRIBUTIONS
1. You toss a coin 8 times. Call heads success. If the coin is fair, the probability of success P is 0.5. What is
the probability of getting exactly 5 heads out of 8 tosses? of exactly 20 heads out of 100 tosses?
2. A bank examiner’s record shows that the probability of an error in a statement for a checking account at
Trust Us Bank is 0.03. The bank statements are sent monthly. What is the probability that exactly two of
the next 12 monthly statements for our account will be in error? Now use the CDF option to find the
probability that at least two of the next 12 statements contain errors. Use this result with subtraction to find
the probability that more than two of the next 12 statements contain errors. You can use the Calculator key
to do the required subtraction.
3. Some tables for the binomial distribution give values only up to 0.5 for the probability of success p. There
is symmetry to the values for p greater than 0.5 with those values of p less than 0.5.
(a) Consider the binomial distribution with n = 10 and p = .75. Since there are 0–10 successes possible,
put 0 – 10 in C1. Use PDF option with C1 and store the distribution probabilities in C2. Name
C2 = ‘P = .75’. We will print the results in part (c).
(b) Now consider the binomial distribution with n = 10 and p = .25. Use PDF option with C1 and store
the distribution probabilities in C3. Name C3 = ‘P = .25’.
(c) Now display C1 C2 C3 and see if you can discover the symmetries of C2 with C3. How does
P(K = 4 successes with p = .75) compare to P(K = 6 successes with p = .25)?
(d) Now consider a binomial distribution with n = 20 and p = .035. Use PDF on the number 5 to get
P(K = 5 successes out of 20 trials with p = .35). Predict how this result will compare to the probability
P(K = 15 successes out of 20 trials with p = .65). Check your prediction by using the PDF on 15 with
the binomial distribution n = 20, p = .65.
The INVCDF command for a binomial distribution can be used in the solution of Quota problems as described
in Section 5.3 of Understandable Statistics.
4. Consider a binomial distribution with n = 15 and p = 0.64. Use the INVCDF to find the smallest number
of successes K for which P(X ≤ K) = 0.98. What is the smallest number of successes K for which
P(X ≤ K) = 0.09?
COMMAND SUMMARY
To Find Probabilities
PDF E [E] calculates probabilities for the specified values of a discrete distribution and
calculates the probability density function for a continuous distribution.
CDF E [E] gives the cumulative distribution. For any value X, CDF X gives the
probability that a random variable with the specified distribution has a value less than or equal to X.
INVCDF E [E] gives the inverse of the CDF.
Each of these commands apply the following distributions (as well as some others). If no subcommand is
used, the default distribution is the standard normal.
BINOMIAL n = K p = K
POISSON
µ = K (note that for the Poisson distribution µ = λ)
INTEGER a = K b = K
DISCRETE values in C, probabilities in C
NORMAL [µ = K [σ = K]]
UNIFORM [a = k b = K]
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Technology Guide Understandable Statistics, 8th Edition
T
d.f. = K
F
d.f. numerator = K d.f. denominator = K
CHISQUARE
d.f. = K
WINDOWS menu selection: hCalchProbability DistributionhSelect distribution
In the dialog box, select Probability for PFD; Cumulative probability for CDF; Inverse cumulative for
INV; Enter the required information such as E, n, p, or µ, d.f. and so forth.
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