M116 – NOTES – CH 5
Section 5.1 – Probability Distributions

Requirements for a Probability Distribution
o
o

0  P(X = x)  1
The sum of the probabilities of a discrete random variable is 1.
 P( X  x)  1
Finding the mean (expected value) and standard deviation for a probability distribution
Here are the formulas to compute the mean and standard deviation for a probability distribution.
  [ x  P( x)]
Mean
Standard Deviation

[( x   )
2
 P( x)]
Where x are the possible values of the random variable, and P(x) are the corresponding
probabilities.

To evaluate the mean and standard deviation using the calculator
Enter x into L1
Enter the probabilities into L2
Press STAT
Arrow right to CALC
Select 1: 1-Var Stats L1,L2
Press ENTER

Identifying Unusual Results with the Range Rule of Thumb
Maximum usual value =
Minimum usual value =

  2
  2
Identifying Unusual Results with Probabilities
Unusually high: x successes among n trials is unusually high if P(x or more) is very
small (such as less than 0.05)
Unusually low: x successes among n trials is unusually low if P( or fewer) is very
small (such as less than 0.05)
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M116 – TI 83/84 CALCULATOR – CH 5
Using the TI-83/84 calculator to find the Mean and Standard Deviation of Probability
Distributions
To evaluate the mean and standard deviation using the calculator
Enter x into L1
Enter the probabilities into L2
Press STAT
Arrow right to CALC
Select 1: 1-Var Stats L1,L2
Press ENTER
1) When randomly selecting jail inmates convicted of DWI (driving while intoxicated), the
probability distribution for the number x of prior DWI sentences is as described in the
accompanying table (based on data from the U.S. Department of Justice).
x
0
1
2
3
P(x)
0.512
0.301
0.138
0.049
a) What is the population? (Who are we talking about?)
b) Describe in words the random variable. (What are we counting?)
c) What are the possible values of the random variable?
d) Verify that the given table is a probability distribution
e) Use the calculator to find the mean and standard deviation of this distribution.
f) Which values are usual and which are unusual, according to
(i) The probability rule?
(ii) The range rule of thumb?
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M116 – NOTES – CH 5
Section 5.2 & 5.3 – Binomial Experiments

Features of a binomial experiment (5.2)
1)
2)
3)
4)
The experiment has a fixed number of trials (n)
The trials must be independent
Each trial has 2 possible outcomes: success (S) and failure (F)
Probabilities remain constant for each trial.
p is the probability of success, and q is the probability of failure
When sampling without replacement, the events can be treated as if they were
independent if the sample size is no more than 5% of the population size. (That
is, n  0.05 N )

Binomial probability formula (5.2)
P( r ) 
n!
 p r  q nr = Cn,r  p r  q nr
(n  r )!r !
where x is the number of successes in n trials, p is the probability of success in
any one trial, and q is the probability of failure in any one trial. (q = 1 – p)

Find binomial probabilities with a shortcut feature of the calculator
To find individual probabilities: Use binompdf(n,p,x)
Press 2nd VARS
Select 0:binompdf(
Type n,p,x)
Press ENTER
To calculate cumulative probabilities from 0 to x, use binomcdf(n,p,x)

Mean, Variance, and Standard Deviation for the Binomial Distribution (5.3)
In Section 5.1, we used the general formulas for any discrete probability distribution. But
the special qualities of binomials distributions, lead to specialized formulas for binomials
  npq
  np

Both sets of formulas form sections 5.1 and 5.3 are given on the table below
Any Discrete Probability
Distribution
Section 5.1
Mean
Standard Deviation
  [ x  P( x)]

[ x
2
 P ( x)]   2
Binomial Distributions
Section 5.3
  np
  npq
3
M116 – TI 83/84 CALCULATOR – CH 5
2) Count the number of male and female students in class, and complete the table.
frequency
Probability
Male
Females
Total
a) EXPERIMENT: We are going to select 4 students at random from the class, and count the
number of females in the selected group.
This is an experiment with 4 trials.
Then
n=4
Since we are counting females, our success probability p, is the probability obtained above for
the females category.
Then
p=
(use 3 decimal places)
b) On the table given below, indicate the possibilities for the random variable, the number of
female students in a group of 4.
x
Tally…………………...
c)
frequency
Experimental probability
Theoretical probability –from part (e)
Let’s use the calculator to do the random selection.
MATH
Arrow right to PRB
Select 5:randInt(1,….,4)
ENTER
d) Use the class roster to identify the gender of the selected students. Count the number of female
students in the group of 4. Tally the results on the table.
Do the process 5 times with your calculator. Come and tally your results on the board.
After calculating the frequencies, determine the experimental probabilities.
e)
Now let’s use the calculator to obtain the theoretical probabilities
STAT
1:Edit
Enter the values of the random variable 0-4 into L1
Go to list 2. Arrow up to “sit on the name of the list”
2nd VARS[DISTR]
Select 0:binompdf(n,p) (use the values of n and p from part (a))
Press ENTER
Compare the experimental and theoretical results. How can you obtain “closer results”?
f)
Observe the table. What values of x are more likely?
g) Use the calculator to find the mean and standard deviation of the probability distribution that you
have in lists 1 and 2 of the calculator.
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M116 – TI 83/84 CALCULATOR – CH 5
h) Identify usual and unusual results, according to
(i) The probability rule?
(ii) The range rule of thumb?
3) Suppose we select 5 students at random from our class. Use the calculator to construct
the theoretical probability distribution for the number of male students in a group of 5. Use
the probability of male obtained on the first table of the preceding page. Place this
distribution into lists 3 and 4 of your calculator.
In this problem n = …………….., and p = ………………..
x
P(x)………………
a) Use the calculator to find the mean and standard deviation of the probability distribution that you
have in lists 3 and 4 of the calculator.
b) Identify usual and unusual results, according to
(i) The probability rule?
(ii) The range rule of thumb?
5
M116 – TI 83/84 CALCULATOR – CH 5
Using the TI-83/84 calculator to find Factorials, Binomial Coefficients and Binomial
Probabilities
4) Evaluate factorials with the calculator:
Type number
Press MATH
Arrow left to PRB
Select 4:!
Press ENTER
Examples:
a) Find 10!
10! = 10 * 9 * 8 *… * 3 * 2 * 1 =
b) Find 6!
5) Evaluate binomial coefficients with the calculator:
Type n (number of trials)
Press MATH
Arrow left to PRB
Select 3:nCr
Type x
Press ENTER
Examples:
a) Find 10 C3 =
b) Find 8 C5 =
6) Evaluate binomial probabilities with the formula:
For a binomial experiment with n = 7 and p = 0.8,
a) Find P(r = 3)
b) Find P(r = 6)
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M116 – TI 83/84 CALCULATOR – CH 5
7) Find binomial probabilities with a shortcut feature of the calculator
A) To find individual probabilities: Use binompdf(n,p,x)
Press 2nd VARS
Select 0:binompdf(
Type n,p,x)
Press ENTER
Examples:
a) For a binomial experiment with n = 7 and p = 0.8, find P(r = 3).
b) For a binomial experiment with n = 4 and p = 1/3, find P(r = 2).
5-B) To get a list of all the probabilities corresponding to x = 0, 1, 2, …., n:
Use binompdf(n,p) and scroll to the right to read the probabilities
Examples:
a) For a binomial experiment with n = 4 and p = 1/6, find the probability
distribution. Find the mean and standard deviation for the distribution. Identify
usual and unusual values with the range rule of thumb and with the probability rule.
X P(X=r)
b) For a binomial experiment with n = 5 and p = 1/2, find the probability distribution.
Find the mean and standard deviation for the distribution. Identify usual and unusual
values with the range rule of thumb and with the probability rule.
X P(X=x)
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M116 – TI 83/84 CALCULATOR – CH 5
5-C) To calculate cumulative probabilities from 0 to x, use binomcdf(n,p,x)
Press 2nd VARS
Select A:binomcdf(
Type n,p,x)
Press ENTER
Examples:
a) For a binomial experiment with n = 7 and p = 0.2, find the probability of at
most 3 successes.
b) For a binomial experiment with n = 6 and p = 0.46, find the probability of at
most 4 successes.
c) For a binomial experiment with n = 4 and p = 0.3, find the probability of at
least 2 successes.
d) For a binomial experiment with n = 8 and p = 0.85, find the probability of at
least 5 successes.
e) For a binomial experiment with n = 9 and p = 0.35, find the P (2 < x <6)
f) For a binomial experiment with n = 10 and p = 0.73, find the probability that x is
between 4 and 9 inclusive.
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M116 – TI 83/84 CALCULATOR – CH 5
OPTIONAL (ITP) - Using the TI-83/84 calculator to graph Binomial Probability
Distributions
6) To graph a probability distribution follow the steps outlined below:
a) For a binomial experiment with n = 4 and p = ¼
Get into the editor of the calculator and clear two lists
Place the possible values of the random variable into one of the lists, let’s say L1
(In this case the possible values of x are from 0 to 4)
Go to L2, and arrow up until you “sit” on the name of the list
Press 2nd VARS[DISTR]
Arrow down to select 0:binompdf(
Indicate choices of n,p (It should read: binompdf(4,1/4))
Press ENTER
The probabilities should show in L2
Sketch a HISTOGRAM for L1, L2 (GO to STAT PLOT)
Select appropriate WINDOW values
x-min = 0
x-max = 5 (n + 1)
x-scale = 1
y-min = - 0.2
y-max = 0.6
Press GRAPH
The graph of the distribution should show.
Press TRACE and arrow right to see the values of the random variable along
with the probabilities.
Comment on the shape of this distribution.
b) For a binomial experiment with n = 6 and p = ½
Sketch the graph of the distribution and comment on its shape. Work on L3, L4.
Remember to have only one plot ON.
c) For a binomial experiment with n = 10 and p = 4/5
Sketch the graph of the distribution and comment on its shape. Work on L5, L6.
d) Relationship between the value of p and the shape of the binomial distribution
If p < 0.5
the shape is
If p = 0.5
the shape is
If p > 0.5
the shape is
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M116 – TI 83/84 CALCULATOR – CH 5
Binomial Distributions and Simulations (Chapter 5)
7) – Booking tickets:
Air America has a policy of booking as many as 15 persons on an airplane that can seat only
14. Past studies have revealed that only 85% of the booked passengers actually arrive for the
flight. Find the probability that if Air America books 15 persons, not enough sits will be
available.
a) Use the formula.
b) Is this probability low enough so that overbooking is not a real concern for passengers?
Is it unusual to find that there are not enough sits available?
c) Now use a feature in the calculator to calculate the answer. Indicate what you enter in
the calculator, and the results.
d) OPTIONAL (ITP)
Now we are going to simulate this situation by repeating the experiment 20 times.
Use MATH PRB 7:randBin(n,p) and press ENTER 20 times.
Record results in a table, and then use your table to answer the question to the problem.
e) Use class results and answer the question again.
f) OPTIONAL (ITP) Here we have another simulation technique. Use the calculator to
generate 50 numbers that come from a binomial distribution with n = 15 and p = 0.85
(We’ll clear List 1, generate the numbers and store them into List 1, we’ll sort
the list and then explore the editor)
STAT 4:ClrList L1 :
MATH PRB 7:randBin(n,p,50) STO L1 :
STAT 3:SortA(L1)
Go to the editor, explore the list and count how many times we had 15 passengers
showing up. Then determine the probability, and compare with the theoretical results
from part (a).
Comment on the law of large numbers.
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M116 – NOTES – CH 6
Normal Distributions (Chapter 6)

Finding probabilities (areas)
Step 1: Draw the graph shading the area desired. Label the mean and the specific xvalues being considered.
Step 2: Find the z-Score for each x-value involved.
Z-score = (score – mean) / standard deviation
Z-SCORE:
z
x

Step 3: Use table 5 to find the cumulative left area bounded by z.
Step 4: Answer the problem.

Using the TI-83/84 to obtain probabilities, percentages, areas
Press
2nd, VARS
Select
2:normalcdf(
Type
left endpoint, right endpoint, , )

Finding Values from Know Areas (Probabilities)
Working BACKWARDS
Step 1: Draw the graph, shade and label the given area, and identify the location
of the x-value being sought.
Step 2: Find the cumulative left area bounded by x.
Step 3: Use table 5 to find the z-score.
(Go backwards! From the main body of the table to the z-score)
Step 4 Find the score, x, by using the formula
x    z

(score = mean + z-score * standard deviation)
Using the TI-83/84 to obtain normal scores, percentiles
Press
2nd, VARS
Select
3:invNorm(
Type
total area to the left of the desired value, , )
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M116 – TI 83/84 CALCULATOR – CH 6
Normal Distributions and Simulation (Section 6.3)
8) - The United States Air Force ACES-II ejection seat used in fighter jets have been originally
designed for men whose weight is between 140 and 211 pounds. Nowadays many women are
joining the air force and we wonder if it is necessary to re-design the ejection sits.
We’ll take into consideration that weights of women are normally distributed with a mean of
143 and a standard deviation of 29 pounds, and that weights of men are normally distributed
with a mean of 170 and a standard deviation of 40 pounds.
(8-i) If a woman is randomly selected, what is the probability that her weight is
between 140 and 211 pounds?
a) Show all your work, along with the diagram and the shading of the area you are
calculating.
b) Use a feature of the calculator to find the answer.
c) OPTIONAL (ITP) Now we’ll simulate the problem by generating 50 numbers that
come from a normal distribution with a mean of 143 and a standard deviation of 29
pounds.
(We’ll clear List 1, generate the numbers and store them into List 1, we’ll sort the
list and then explore the editor)
STAT 4:ClrList L1 :
MATH PRB 6:randNorm(mean,st-dev,50) STO L1 :
STAT 3:SortA(L1)
Go to the editor, explore the list and count how many women have weights in the given
interval. Then determine the probability.
How does the experimental probability compare with the theoretical probability from
part (a)? Comment on the law of large numbers.
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M116 – TI 83/84 CALCULATOR – CH 6
(8-ii) If a man is randomly selected, what is the probability that his weight is between 140
and 211 pounds?
a) Show all your work, along with the diagram and the shading of the area you are
calculating.
b) Use a feature of the calculator to find the answer.
c) OPTIONAL (ITP) Now we’ll simulate the problem by generating 50 numbers that
come from a normal distribution with a mean of 170 and a standard deviation of 40
pounds.
(We’ll clear List 2, generate the numbers and store them into List 2, we’ll sort the
list and then explore the editor)
STAT 4:ClrList L2 :
MATH PRB 6:randNorm(mean,st-dev,50) STO L2 :
STAT 3:SortA(L2)
Go to the editor, explore the list and count how many men have weights in the given
interval. Then determine the probability.
How does the experimental probability compare with the theoretical probability of part
(a)? Comment on the law of large numbers.
d) Are women at greater risk of being injured? Do you think it is necessary to redesign the
ejection seats?
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M116 – TI 83/84 CALCULATOR – CH 6
9) Designing Helmets
Engineers must consider the breadths of male heads when designing motorcycle helmets.
Men have head breaths that are normally distributed with a mean of 6.0 in. and a standard
deviation of 1.0 in. Due to financial constraints, the helmets will be designed to fit all men
except those with head breaths that are in the smallest 2.5% or largest 2.5 %. Find the
minimum and maximum head breaths that will fit men.
a) Show all steps to get the answer.
b) Now use a feature of the calculator to answer.
c) OPTIONAL (ITP) We are going to use simulation to find the probability that a man
selected at random has a head breath smaller than the MINIMUM found in part (a).
Generate 50 numbers that come from a normal distribution with a mean of 6 and a
standard deviation of 1.
STAT 4:ClrList L3 :
MATH PRB 6:randNorm(mean,st-dev,50) STO L3 :
STAT 3:SortA(L3)
Explore the list to count how many men in the sample have head breaths smaller than
the minimum found in part (a). What percent of the sample is this? How does it
compare to 2.5%?
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M116 – NOTES – CH 6
Normal Approximation to Binomial Distributions (Section 6.4)
When can we use the Normal distribution to approximate a Binomial distribution?

Normal Distributions as Approximation to Binomial Distributions
If
np  5 and nq  5 , then the binomial random variable has a probability distribution
that can be approximated by a normal distribution with the mean and standard deviation
given as
  n* p
  n* p*q
10) For each of the following problems indicate whether the normal distribution is appropriate
to approximate the binomial distribution. If so, calculate the probability of exactly 5 successes
using the binomial distribution and then estimate the probability using the normal distribution.
a) Case I: n = 12, p = 0.5
b) Case II: n = 12, p = 0.2
c) Case III: n = 12, p = 0.9
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M116 – NOTES – CH 6
Summary

Identifying Unusual Results with the Range Rule of Thumb
According to the range rule of thumb, most values should lie within 2 standard
deviations from the mean. We can therefore identify “unusual” values by determining if
they lie outside these limits:
Minimum usual value = μ - 2σ
Maximum usual value = μ + 2σ

Identifying Unusual Results with Probabilities
Unusually high: x successes among n trials is an unusually high number of successes
if
P(x or more) is very small (such as 0.05 or less).
Unusually low: x successes among n trials is an unusually low number of successes if
P(x or fewer) is very small (such as 0.05 or less).

Rare Event Rule
If, under a given assumption the probability of a particular observed event is extremely
small, we conclude that the assumption is probably not correct.

Normal Distributions as Approximation to Binomial Distributions
If
np  5 and nq  5 , then the binomial random variable has a probability distribution
that can be approximated by a normal distribution with the mean and standard deviation
given as
  n* p
  n* p*q
16
CHAPTER 6
11) Singular is a medication whose purpose is to control asthma attacks. In clinical trials of
Singular, 18.4% of the patients in the study experienced headaches as a side effect.
a) Compute the mean and the standard deviation of a random variable X, the number of
patients
experiencing headaches in 400 trials of the probability experiment.
b) Would it be unusual to observe 86 patients who experience headaches in a random
sample of 400 patients who use this medication? Explain why for each of the following.
(i) According to the range rule of thumb
(ii) To use the probability rule, calculate P(x ≥86)
 Use methods of chapter 5 to calculate the probability.

Verify that the normal distribution is appropriate to estimate this
probability.

Estimate the probability by using the normal distribution. Use the
continuity correction factor. (Note: for large n, the continuity
correction factor may not be necessary)
Show steps, and then check with calculator feature. Remember to
answer the question to the problem.
17
CHAPTER 6
c) Would it be unusual to observe 93 patients who experience headaches in a random
sample of 400 patients who use this medication? Explain why for each of the following.
(i) According to the range rule of thumb
(ii) To use the probability rule, calculate P(x ≥ 93)
 Use methods of chapter 5 to calculate the probability.

Verify that the normal distribution is appropriate to estimate this
probability.

Estimate the probability by using the normal distribution. Use the
continuity correction factor. (Note: for large n, the continuity
correction factor may not be necessary)
Show steps, and then check with calculator feature. Remember to
answer the question to the problem.
d) The probability that in groups of 400 patients, 93 or more experience headache as a
side effect is _______ This means, in 1000 trials of this experiment we expect about
________ trials to result in 93 or more experiencing headaches. Because this event only
happens _______ out of __________ times, we consider it to be usual/unusual
e) If in a random sample of 400 patients who use this medication you actually observe
93 patients who experience headaches; what might you conclude about the actual
percentage of patients who experience headaches? (Read rare event rule on page 13)
18
CHAPTER 6
12) Depakote is a medication whose purpose is to reduce the pain associated with migraine
headaches. In clinical trials and extended studies of Depakote, 2% of the patients in the study
experienced weight gain as a side effect.
a) Compute the mean and standard deviation of the random variable X, the number of
patients
experiencing weight gain in 600 trials of the probability experiment.
b) Would it be unusual to observe 16 or more patients who experience weight gain in a
random sample of 600 patients who take the medication? Explain why for each of the
following.
(i) According to the range rule of thumb
(ii) To use the probability rule, calculate P(x ≥ 16)
 Use methods of chapter 5 to calculate the probability.

Verify that the normal distribution is appropriate to estimate this
probability.

Estimate the probability by using the normal distribution. Use the
continuity correction factor. (Note: for large n, the continuity
correction factor may not be necessary)
Show steps, and then check with calculator feature. Remember to
answer the question to the problem.
19
CHAPTER 6
c) Would it be unusual to observe 21 patients who experience weight gain in a random
sample of 600 patients who take the medication? Explain why for each of the following.
(i) According to the range rule of thumb
(ii) To use the probability rule, calculate P(x ≥ 21)
 Use methods of chapter 5 to calculate the probability.

Verify that the normal distribution is appropriate to estimate this
probability.

Estimate the probability by using the normal distribution. Use the
continuity correction factor. (Note: for large n, the continuity
correction factor may not be necessary)
Show steps, and then check with calculator feature. Remember to
answer the question to the problem.
d) The probability that in groups of 600 patients, 21 or more experience weight gain as
a side effect is _______ This means, in 1000 trials of this experiment we expect about
________ trials to result in 21 or more experiencing weight gain. Because this event
only happens_______ out of __________ times, we consider it to be usual/unusual
e) If in a random sample of 600 patients who use this medication you actually observe
21 patients who experience weight gain; what might you conclude about the actual
percentage of patients who experience weigh gain? (Read rare event rule on page 13)
20