5 – Binomial Random Variable and Probability Distribution
5.1 - Motivating Example and Definitions:
5.2 - Binomial Distribution and Random Variable
A binomial random variable, X, is defined to be the number of “successes” in n
independent trials where the P(success) = p is constant for each of trials. In the definition
above notice the following conditions need to be satisfied for a binomial experiment:
1. There is a fixed number of n trials carried out.
2. The outcome of a given trial is either a “success” or “failure”.
3. The probability of success (p) remains constant from trial to trial.
P( success )  p
and P( failure )  (1  p) = q
4. The trials are independent, that is the outcome of a given trial is not affected by the
outcome of any other trial.
Binomial Probability Function
n
n!
P( X  x)    p x q n  x 
p x q n  x , x  0,1,..., n
x
x
!
(
n

x
)!
 
where
k! 1  2  3    k
“k factorial” is the product of the numbers 1 through k and is denoted by k!
The factorial expression in front counts the number of ways it is possible to obtain x
successes in n trials while the remainder of the formula gives the probability of getting a
particular sequence with x “successes” and n – x “failures”.
Mean # of successes  = np
Standard deviation of the # of successes = √𝑛𝑝𝑞
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The JMP spreadsheet Binomial Table Generator does all the probability calculations for
the following:
 P(X = x) = probability of obtaining x “success” in n trials.
 P(X < x) = probability of obtaining x or less “successes” in n trials.
 P(X > x) = probability of obtaining x or more “successes” in n trials.
To use the Binomial Table Generator file in JMP you simply need to change the number
of independent trials (n) and the probability of success (p) in the table below. To change
the n and p values right-click at the top of the column and select Formula. You can then
change the number in the formula to your desired values for n and p. The table
automatically updates and gives the following probabilities in the last three columns. Be
sure to look at the comments on each in the upper left-hand corner of the table.
Example 1: Drug Side Effects
A drug company claims that 10% of patients taking the drug will experience adverse side
effects. To test this claim a researcher administers the drug to a random sample of 20
patients. Let X = the number patients in our sample who experience adverse side effects.
Does this study meet the conditions of binomial probability experiment?
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To set up the binomial probability table for this situation we need to change the number
of trials to n = 20 and the probability of success to p = .10, to do this right-click on the
top of each column and select Formula.
a) What is the probability that exactly 4 patients experience side effects?
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b) What is the probability that 2 patients or less experience side effects?
c) What is the probability that 5 patients or less experience side effects?
d) What is the probability that 6 or more patients experience side effects?
e) Suppose that in our sample 6 patients experience side effects. What do think about the
claim made by the drug company on the basis of this result?
f) Suppose we gave the drug to 100 patients. If the drug company’s claim regarding side
effects is true, how many side effects do we expect to observe, i.e. what is the mean or
expected value of X? Recall: For a binomial random variable   np
g) Find the standard deviation of the number of side effects and use it to give an interval
that is very likely to cover the number of side effects observed when giving the drug to
n = 100 patients. Recall: For a binomial random variable   npq
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When n is sufficiently large then X = # of “successes” in the n trials is approximately
normally distributed. As an example consider the histogram below which shows the
simulated results of 10,000 clinical trials where for each trial n = 100 patients are given a
drug which has p  P( side effect)  .10 or a 10% chance of causing a side effect and the
number of patients experiencing side effects is observed.
h) How many side effects would you have to observe to be convinced that the 10% side
effect is wrong and that the true side effect rate is greater?
Binomial Distribution Table for n = 100 and p = .10
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Example 2: Effect of Togetherness on Heart Rates of Rats (SIGN TEST)
A researcher was interested in determining if the heart rate of rats increases when they
are in a cage with other rats versus when they are in a cage by themselves. The
researcher thought this would be the case but wanted to conduct a study to determine if
this hypothesis was supported empirically.
The results were obtained:
Rat Alone Together Difference Sign of Difference
(A)
(T)
=T-A
1
463
523
2
462
494
3
462
461
4
456
535
79
5
450
476
26
6
426
454
28
7
418
448
30
8
415
408
-7
9
409
470
61
10
402
437
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What did we conclude regarding the research hypothesis?
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Example 3: Swain vs. Alabama
In 1965, an appeal to the Supreme Court was made by a black man (the petitioner)
sentenced to death after being convicted in the Circuit Court of Talladega County,
Alabama, of the rape of a 17 year-old white girl by an all-white jury. At the time of the
crime, the defendant was 19. The petitioner alleged (among other things) that the entire
process of selecting eligible jurors, from the jury pool of eligible jurors to the jury panel
(from which the trial jury is chosen), was racially discriminatory.
At the time, in the early 1960’s, eligible jurors in Alabama were males over 21.
According to Census figures available then, there were 16,406 individuals that fit the
profile in Talladega Country. Twenty-six percent of these individuals were AfricanAmericans. When 100 individuals were chosen to serve on the jury panel, only 8 of them
were African-Americans*. What is the likelihood of getting 8 blacks in a sample of 100
drawn from the population of males eligible for jury duty in Talladega County?
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