Download File - Different Uses for Labs

MAT 155 Lab
Binomial Distribution – TI83, TI84
The binomial distribution is a probability distribution of a discrete random variable (data is countable).
The features of a binomial experiment include: Having a fixed number of trials (n). The n trials are
independent and repeated under identical conditions. Each trial has only 2 outcomes of Success or
Failure. The probability of success p is the same for each individual trial. The probability of failure is
equal to 1 – p. The random variable r is used to count the number of successes out of n trials.
Complete Parts I and IV using the answer key posted in Moodle. Work through parts II and III
as an example of how to use the calculator to complete these exercises.
I.
Prerequisite Probability Notation
You must know how to use probability notation before you can calculate binomial probabilities
correctly on the calculator. The following are examples of how to do this.
i.e. The probability of 2 successes. This means we must calculate the probability of exactly 2
successes. The statement would be P(X = 2). There is no other way to calculate it.
i.e. The probability of at least 4 successes. This means we must calculate the probability of 4
successes and all probabilities more than 4. The statement would be P(X ≥ 4). The probability
could also be calculated using 1 – P(X ≤ 3), which is what you would have to use to calculate it on
your calculator.
i.e. The probability of 3 or fewer successes. This means we must calculate the probability of 3
successes and all probabilities less than 3. The statement would be P(X ≤ 3).
For the following problems, create an appropriate probability statement using probability notation
and show how to calculate the same probability using a statement with ≤. The calculator only
calculates binomials with a ≤, so you have to know how to do this.
1.
2.
3.
4.
5.
The probability of exactly 6 successes.
The probability of at least 10 successes.
The probability of less than 7 successes.
The probability of at most 7 successes.
The probability of no fewer than 4
successes.
6. The probability of 2 or more successes.
7. The probability of 4 or fewer successes.
8. The probability the number of successes
does not exceed 3.
9. The probability of fewer than 9
successes.
10. The probability the number of successes
exceeds 2
STOP HERE – Try to answer questions 1 – 10 on the answer key in Moodle before continuing.
You can try this multiple times until you understand what you are doing before you go on to
actual binomial calculations.
1
Using the calculator -- There are two main functions on the calculator that calculate probabilities
from a binomial distribution: binompdf( ) and
binomcdf( ).
II.
Computing the probability of x successes for an EXACT number of trials. binompdf( )
Example: Privacy is a concern for many users of the Internet. One survey showed that 59% of
Internet users are somewhat concerned about the confidentiality of their email. Based on this
information, what is the probability that for a random sample of 10 Internet users, 6 are concerned
about the privacy of their email?
(Understandable Statistics, Brase & Brase, 8th Edition)
1. First you must know n, p, x.
n = 10
p = 59% = .59
x=6
where p represents the probability Internet users are concerned about confidentiality.
2. Now we are ready to use the calculator. For this example, it is important to note that the
question is asking for P(X = 6).
 Press 2nd VARS on your calculator to get to the DISTR menu.
 Now scroll down and press ENTER on binompdf(
 In the parentheses, you will type the number of trials n, the probability of success p,
the number of successes out of n trials x.
 So for this example, you will have binompdf(10,.59,6) and press ENTER.
3. Your answer should be .25. This means there is a 25% chance that EXACTLY 6 of the 10
Internet users are concerned about the privacy of email.
III.
Computing the probability of a range of successes. binomcdf( )
Example: A biologist is studying a new hybrid tomato. It is known that the seeds of this hybrid
tomato have probability 0.70 of germinating. The biologist plants six seeds.
(Understandable Statistics, Brase & Brase, 8th Edition)
1. What is the probability that 2 or fewer seeds will germinate?
a. First you must know n and p.
n=6
p = 0.70
where n represents number of seeds and p represents the probability of germinating
b. Now we are ready to use the calculator. For this example, it is important to note that
the question is asking for P(X ≤ 2).
 The calculator will only calculate the range below, so the probability must always
be converted to a less than or equal to, in this example it is not necessary.
 Press 2nd VARS on your calculator to get to the DISTR menu.
 Now scroll down and press ENTER on binomcdf(
- The c stands for cumulative.
 In the parentheses, you will type the number of trials n, the probability of success p,
the number of successes out of n trials x.
 So for this example, you will have binomcdf(6,.7,2) and press ENTER.
c. Your answer should be .07. This means there is a 7% chance that 2 or fewer seeds
will germinate out of the 6 seeds planted.
2
2. What is the probability that at least four seeds will germinate?
a. First you must know n, p, x.
n=6
p = 0.70
where n represents number of seeds and p represents the probability of germinating
b. Now we are ready to use the calculator. For this example, it is important to note that
the question is asking for P(X ≥ 4). (“at least” means it must be the given number
and anything higher)
 You must also know the following: P(X ≥ 4) = 1 – P(X ≤ 3). We can use 1 – the
probability of x less than or equal to three because we are using discrete data.
Meaning there are no decimal values in between 3 and 4 since the data is discrete.
 The calculator will only calculate the range below, so the probability must always
be converted to a less than or equal to.
 Note that x is now 3 instead of 4.
 Press 2nd VARS on your calculator to get to the DISTR menu.
 Now scroll down and press ENTER on binomcdf(
- The c stands for cumulative.
 In the parentheses, you will type the number of trials n, the probability of success p,
the number of successes out of n trials x.
 So for this example, you will have binomcdf(6,.7,3) and press ENTER.
 This is not the correct answer. This answer is the P(X ≤ 3). So calculate 1 – the
given probability to get your answer. To do this you can press 1 – 2nd (-) to use the
previous answer in your calculation.
c. Your answer should be .74. This means there is a 74% chance that at least 4 seeds
will germinate out of the 6 seeds planted.
3. What is the probability that between 3 and 5 seeds will germinate?
a. First you must know n, p, x.
n=6
p = 0.70
where n represents number of seeds and p represents the probability of germinating
b. Now we are ready to use the calculator. For this example, it is important to note that
the question is asking for P(3 ≤ X ≤ 5).
 You must know that the statement above can be calculated using the following
statements: P(3 ≤ X ≤ 5) =
this can be calculated in 2 different ways
One Way: = P(X = 3) + P(X = 4) + P(X = 5)
= binompdf(6, .7, 3) + binompdf(6, .7, 4) + binompdf(6, .7, 5) = .812

Other Way:= P(X ≤ 5) – P(X ≤ 2)
= bindomcdf(6, .7, 5) – binomcdf(6, .7, 2) = .812
The reason that the last binomcdf must use 2 is because the numbers are discrete so
you are trying to calculate the probability of between 3 and 5 – This includes the
numbers 3, 4, and 5.
In order to do this you find the probability of the numbers 1, 2, 3, 4, and 5. Then you
subtract of the numbers 1 and 2 to be left with the probability of 3, 4, and 5.
c. Your answer should be .812. This means there is an 81.2% chance that between 3
and 5 seeds will germinate out of the 6 seeds planted.
3
IV.
Follow-Up Questions
Use previous information as an example for the following problems.
1. Follow the instructions given to complete the problem.
According to the Information Please Almanac, 80% of adult smokers started smoking before
turning 18 years old. Suppose 10 smokers 18 years old or older are randomly selected and
the number of smokers who started smoking before 18 is recorded.
You must know the following from reading the problem:
n = 10
p = .8
where p represents the probability adult smokers started smoking before turning 18 years old
a. Is this a binomial experiment? (This will be a T/F question on the answer key)
b. Find the probability that exactly 8 of them started smoking before 18 years of age.
c. Find the probability that at least 8 of them started smoking before 18 years of age.
d. Find the probability that fewer than 8 of them started smoking before 18 years of
age.
e. Find the probability that between 7 and 9 of them, inclusive, started smoking before
18 years of age.
Calculate the following on your own using previous information as an example.
2. (# 38 p. 341 from book) Allergy Sufferers.
Clarinex-D is a medication whose purpose is to reduce the symptoms associated with a
variety of allergies. In clinical trials of Clarinex-D, 5% of the patients in the study
experienced insomnia as a side effect. Suppose a random sample of 20 Clarinex-D users is
obtained and the number of patients who experienced insomnia is recorded.
a. Find the probability that exactly 3 experienced insomnia as a side effect.
b. Find the probability that 3 or fewer experienced insomnia as a side effect.
c. Find the probability that 3 or more experienced insomnia as a side effect.
d. Find the probability that between 1 and 4 patients, inclusive, experienced insomnia
as a side effect.
e. Would it be unusual to find 4 or more patients who experienced insomnia as a side
effect? (This will be a T/F question on the answer key)
Finish the answer key in Moodle to receive a grade for this lab.
4