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Chapter 4
• Discrete Probability Distributions
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
DISCRETE PROBABILITY DISTRIBUTIONS
SECTION 4-2
I. BINOMIAL PROBABILITIES
In the previous chapter, we looked at statistical experiments and computed the
probabilities of specified events. We will now examine a particular type of statistical
experiment called a binomial experiment.
A. Characteristics of a binomial experiment.
1. The same action is repeated
a) Conditions for repetition must be identical.
b) One trial must be independent of all others. (the results of one trial
cannot affect another.)
c) The number of trials is n.
2. A binomial experiment must have exactly two outcomes.
a) Success (defined in the problem)
b) Failure (all outcomes that do not qualify as successes.)
3. On an individual trial
a) P (success) = p
b) P (failure) = q
c) q = 1 – p (They are complementary events)
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
DISCRETE PROBABILITY DISTRIBUTIONS
SECTION 4-2
4. The number of trials that are successful is denoted r where r ≤ n.
B. The basic problem is to find the probability of getting exactly r successes out
of n trials
1. In a binomial experiment we use r as the random variable.
a) r counts the number of successful outcomes.
b) r is a discrete random variable.
c) For each experiment, the possible values of r can be listed, r = 0, 1,
2, 3, … n. The values range from r = 0 (no successful outcomes) to r
= n (all outcomes are successful).
2. The probability of getting exactly 0 successes (all failures) is denoted P(0),
the probability of getting exactly 1 success is denoted P(1), and so on.
a) Since the categories of no successes, exactly one success, exactly
two successes, etc. are mutually exclusive events, we can add their
probabilities to answer a question using the OR combination.
b) The sum of all probabilities = 1. P(0) + P(1) + P(2) + P(3) + .. P(n) = 1.
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
DISCRETE PROBABILITY DISTRIBUTIONS
SECTION 4-2
C. Methods of determining the probability of r successes out of n trials.
1. Formula
a) P(r) = Cn,r pr qn-r
2. Table of Binomial Probabilities found in Appendix II of your text.
3. The TI-84 can solve a binomial distribution problem for us, so that’s the
way we are going to go.
a) Press 2nd VARS (DISTR).
b) This takes you to a screen that lists 16 different types of statistical
distributions. We will be returning to this screen A LOT!!
1) You are looking for the 11th and/or 12th type of distribution
listed.
a). Pressing the Alpha and A keys takes you to the
binompdf function.
1. binompdf gives you the probability that you will
receive PRECISELY (exactly) r successes.
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
DISCRETE PROBABILITY DISTRIBUTIONS
SECTION 4-2
b) Pressing the Alpha and B keys takes you to the binomcdf
function.
1. binomcdf gives you the CUMULATIVE probability for up
to r successes.
c) Once you have selected the type of distribution you want,
either binompdf or binomcdf, you enter the parameters in
identical fashion.
1. The calculator screen will prompt you for trials, p, and
x.
a. enter the values asked for, highlight Paste, and
press Enter.
b. Press the Enter key again to receive your answer.
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
DISCRETE PROBABILITY DISTRIBUTIONS
SECTION 4-2
D. THE MEAN AND STANDARD DEVIATION OF THE BINOMIAL DISTRIBUTION
1. The binomial distribution’s special properties allow for an easier formula
to compute the expected value (mean), the variance, and the standard
deviation.
a. To compute the mean of a binomial distribution, simply multiply
the number of trials by the probability of success on each trial.
Use μ = 𝑛𝑝
b. The variance of a binomial distribution is found by 𝜎 2 = 𝑛𝑝𝑞.
The standard deviation of a binomial distribution is σ = 𝑛𝑝𝑞
2. Application of the binomial distribution
a. Useful in determining the probability of meeting certain quotas or
specifications under success/failure conditions.
b. For these problems, you’ll be determining the value of n (the
number of trials) necessary to meet the given quota.
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
DISCRETE PROBABILITY DISTRIBUTIONS
SECTION 4-2 EXAMPLE
You are taking a 10 question multiple choice quiz with 5 answer choices for each
question. You were absent for the past week and have no idea how to answer any of
the questions, so you guess on all of them.
a) What is the probability that you get all ten questions correct and score a 100?
b) What is the probability that you guess incorrectly on all 10 questions and get a 0?
c) What is the probability that you get at least 1 question correct?
d) What is the probability that you get at least half of the questions correct?
A. n = 10 (number of questions)
p = 1/5, or .20 (probability of guessing the correct answer on one question)
r = 10 for part a, 0 for part b, is greater than or equal to 1 for part c, and greater
than or equal to 5 for part d.
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
DISCRETE PROBABILITY DISTRIBUTIONS
SECTION 4-2 EXAMPLE
B. Let’s do part a.
1. This is asking us for the probability of getting PRECISELY 10 correct, so we
use the binompdf.
2. 2nd VARS Alpha Math (A) gives us binompdf.
You can also just scroll until you see binompdf in the list.
3. Fill in the values 10,.2,10, highlight Paste and press Enter.
a. Your screen should have binompdf(10,.2,10) showing.
4. Press Enter. The screen will tell you that the probability of getting all ten
questions correct is 1.024 x 10-7.
That means that you have a 0.0000001024 chance of getting all 10
correct.
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
DISCRETE PROBABILITY DISTRIBUTIONS
SECTION 4-2 EXAMPLE
C. Part b asks for the probability of getting none of them correct.
1. This is also asking for a precise probability, so we again go to binompdf.
2. This time, we enter 10,.2,0 into the calculator and get the answer of
.1073741824.
That means that you have a 10.74% chance of getting a zero on that quiz.
D. Part c asks for the probability of getting at least one right.
This is a CUMULATIVE probability, as we need to add the probability of 1 to the
probability of 2, all the way up to the probability of 10.
1. The smart way to do this is to realize that the complement of “at least
1” is “0”. Since we have already found P(0) to be .107, we can subtract
that from 1 and get the answer of .893
E. Part d asks for the cumulative probabilities for r = 5,6,7,8,9, and 10. The
calculator can’t do that, but it can find the probability of r = 0,1,2,3, and 4. This
is the complement of 5 and up, so we can subtract this from 1 to get our
answer.
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
DISCRETE PROBABILITY DISTRIBUTIONS
SECTION 4-2 EXAMPLE
1. binomcdf(10,.2,4) gives the cumulative probabilities for r = 0,1,2,3, and
4.
This number is .9672.
When this is subtracted from 1, we get .0328.
So, you have about a 3% chance of getting at least a 50 on that quiz.
Maybe we should try studying!!
F. The calculator will ALWAYS give you the probability of less than or equal to the
number you enter for x.
If you want greater than x, you need to use the complement rule.
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
SECTION 4-2 EXAMPLES
For each of the following:
State whether or not it is a binomial experiment.
If it is a binomial experiment, describe “success” and “failure”.
Identify values for n, q, and the range of values for r.
U.S.A. Today reported on July 27, 1990, that 70% of the people questioned said that
they watched less T.V. than they did a year ago, 22% said they watched the same
amount, and 8% said they watched more. Find the probability that exactly 3 out of a
randomly selected group of 5 will say they watch less T.V. this year than last.
This a binomial experiment, with success being “watch less T.V. this year than last”
and failure being “watch the same or more T.V. than last year”.
n=5
p = .70 (probability that one person will say they watch less T.V. this year)
q = .30 (1-.7)
r = 3 (number of successes)
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
SECTION 4-2 EXAMPLES
For each of the following:
State whether or not it is a binomial experiment.
If it is a binomial experiment, describe “success” and “failure”.
Identify values for n, q, and the range of values for r.
There are 20 M&M candies in a dish. 8 are brown, 3 are red, 5 are green and 4 are
yellow. Two candies are picked from the dish at random. What is the probability that
both are red?
This is not a binomial experiment, since the trials are not independent.
The probability that the first one is red is 3/20.
The probability of the second being red, given that the first one was, is 2/19.
(3/20)*(2/19) = 6/380 = 3/190 = .0158
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
SECTION 4-2 EXAMPLES
For each of the following:
State whether or not it is a binomial experiment.
If it is a binomial experiment, describe “success” and “failure”.
Identify values for n, q, and the range of values for r.
A ten question multiple choice test is given. Each question has four choices. You did not
study and have no clue as to any of the answers, so you have to randomly guess each
answer. What is the probability you guess exactly 6 correctly (and pass)?
This is a binomial experiment.
Success is “guessing the answer correctly” and failure is “guessing incorrectly”
n = 10 (the number of questions)
p = .25 (since there is one chance in four of guessing each answer correctly)
q = .75 (1 - .25)
r=6
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
SECTION 4-2 EXAMPLES
For each of the following:
State whether or not it is a binomial experiment.
If it is a binomial experiment, describe “success” and “failure”.
Identify values for n, q, and the range of values for r.
U.S.A. Today reported on July 27, 1990, that 70% of the people questioned said that
they watched less T.V. than they did a year ago, 22% said they watch the same
amount, and 8% said they watch more.
Find the probability that exactly 3 out of a randomly selected group of 5 will say they
watch less T.V. this year than last.
On the calculator, use binompdf, since we want exactly 3 successes.
2nd VARS binompdf, then enter 5, .7, and 3.
Paste and Enter will give you .3087, which rounds to .309.)
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
SECTION 4-2 EXAMPLES
For each of the following:
State whether or not it is a binomial experiment.
If it is a binomial experiment, describe “success” and “failure”.
Identify values for n, q, and the range of values for r.
U.S.A. Today reported on July 27, 1990, that 70% of the people questioned said that
they watched less T.V. than they did a year ago, 22% said they watch the same
amount, and 8% said they watch more.
Find the probability that between 2 and 4 (inclusive) people out of 5 will say they
watched less T.V. this year than last year.
Use the binompdf function to find P(2), P(3), and P(4).
2nd VARS binompdf (5, .7, 2), (5, .7, 3) and (5, .7, 4)
P(2) = .132
P(3) = .309
P(4) = .360
P(between 2 and 4 inclusive) = P(2) + P(3) + P(4) = .132 + .309 + .360 = .801
PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
SECTION 4-2 EXAMPLES
For each of the following:
State whether or not it is a binomial experiment.
If it is a binomial experiment, describe “success” and “failure”.
Identify values for n, q, and the range of values for r.
U.S.A. Today reported on July 27, 1990, that 70% of the people questioned said that
they watched less T.V. than they did a year ago, 22% said they watch the same
amount, and 8% said they watch more.
Find the probability that at least one will say they watched less T.V. this year than last
year.
P(at least one) could be computed by adding the probabilities for r = 1, 2, 3, 4, and 5.
However, since r =0 is the complement of “at least one”, it is easier to use the
complement rule.
Compute P(at least one) by 1- P(0).
P(0) can be found by using the binompdf distribution, and is .002.
P(at least one) = 1 - .002 = .998.
Assignments:
Classwork: Pages 215-216 #1-14 All
Homework:Pages 216-219 #16-32 Evens