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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?
We are selecting our sample out of jail inmates convicted of DWI
b) Describe in words the random variable. (What are we counting?)
We are counting the number x of prior DWI sentences
c) What are the possible values of the random variable?
0, 1, 2, 3
d) Verify that the given table is a probability distribution
- All probabilities are positive numbers less than or equal to 1
- The sum of the probabilities is 1
e) Use the calculator to find the mean and standard deviation of this distribution.
μ = .7
σ = .9
f) Which values are usual and which are unusual, according to
(i) The probability rule?
3 is unusual because its probability is less than 0.05
We observe about 49 out of 1000 (less than 5 in 100) with 3 prior DWI sentences
(ii) The range rule of thumb?
[μ-2σ, μ + 2σ] = [.7 – 2(.9), .7 + 2(.9)] = [-1.1, 2.5]
Since the numbers 0, 1, and 2 are inside this interval, then, those are usual outcomes.
Since the number 3 is outside of this interval, then 3 is unusual
It’s common for jail inmates convicted of DWI to have anywhere from 0 to 2 prior DWI
sentences. It’s unusual for them to have 3 prior DWI sentences. (Page 3 of notes)
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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) (not done in the Spring 08 – used calculator)
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
  [ x  P( x)]
Binomial Distributions
Section 5.3
  np
Did not use, used calculator
Standard Deviation

[ x
2
 P( x)]   2
  npq
Did not use, used calculator
(Page 4 of notes)
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DID NOT DO THIS ON THE SPRING 2008
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.
(Page 5 of notes)
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M116 – TI 83/84 CALCULATOR – CH 5
h) Identify usual and unusual results, according to
(i) The probability rule?
DID NOT DO THIS ON SPRING 2008
(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?
(Page 6 of notes)
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DID NOT DO THIS ON THE SPRING 2008
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)
(Page 7 of notes)
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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
Partial solutions given - Please perform the final calculations
Examples:
a) For a binomial experiment with n = 7 and p = 0.8, find P(r = 3).
Binompdf(7,0.8,3) =
b) For a binomial experiment with n = 4 and p = 1/3, find P(r = 2).
Binompdf(4,1/3,2) =
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)=binompdf(4,1/6)
0 .48225
1 .3858
2 .11574
3 .01543
4 .00077
Did you notice that this distribution is right skewed?????(High probabilities on
the low end) The lower numbers of successes are MORE COMMON. Because of the
small probability of success (1/4) it’s more common to have a low number of
successes. And this is what the probability rule tells you. 0, 1, or 2 successes are
usual, while 3 or 4 successes are unusual
Now let’s use the range rule of thumb:
1
  n * p  4*  .666667  .7
6
[
1 5
  n * p * q  4* *  .745  .7
6 6
Usual range: [.7-2*.7, .7+2*.7] = [-.7, 2.1]
Usual outcomes: 0, 1, 2
Unusual outcomes: 3, 4
Think of this example as rolling a fair die 4 times and counting the number of
ones obtained. If you have a die, try it. Do the experiment (of rolling a die 4 times
and counting the number of ones obtained in the 4 rolls) many times. You will see
that it’s “common” to obtain 0, 1, or 2 ones. It’s not so common to obtain 3 or 4
ones.
(Page 8 of notes)
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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
0
1
2
3
4
5
P(X=r)=binompdf(5,1/2)
.03125
.15625
.3125
.3125
.15625
.03125
Did you notice the bell shape in this distribution?
According to the probability rule x = 0 and x = 5 are unusual outcomes
Now let’s use the range rule of thumb:
1
  n * p  5*  2.5
2
[
1 1
  n * p * q  5* *  1.1
2 2
Usual range: [2.5-2*1.1, 2.5 + 2*1.1] = [.3, 4.7]
Usual outcomes: 1, 2, 3, 4
Unusual outcomes: 0, 5 (it agrees with the probability rule)
Think of this example as tossing a fair coin 5 times and counting the number of
heads obtained. If you have a coin, try it. Do the experiment (tossing the coin 5
times and counting the number of heads obtained) many times. You will see that
it’s “common” to obtain 1, 2, 3, or 4 heads in 5 tosses, and not so common to obtain
0 heads or 5 heads.
(Page 8 of notes-continued)
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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
Partial solutions given - Please perform the final calculations
Examples:
a) For a binomial experiment with n = 7 and p = 0.2, find the probability of at
most 3 successes.
P(x ≤ 3) = P(x=0) + P(x=1) + P(x=2) + P(x=3) = binomcdf (7, .2, 3) =
b) For a binomial experiment with n = 6 and p = 0.46, find the probability of at
most 4 successes.
P(x ≤ 4) = binomcdf (6, 0.46, 4) =
c) For a binomial experiment with n = 4 and p = 0.3, find the probability of at
least 2 successes. (This is the complement of at most 1)
P(x ≥ 2) = from the WHOLE probability distribution, subtract the sum of the
probabilities from 0 through 1) =
= 1 - binomcdf (4, 0.3, 1)) =
d) For a binomial experiment with n = 8 and p = 0.85, find the probability of at
least 5 successes. (This is the complement of at most 4)
P(x ≥ 5) = from the WHOLE probability distribution, subtract the sum of the
probabilities from 0 through 4) = 1 - binomcdf (8, 0.85, 4) =
e) For a binomial experiment with n = 9 and p = 0.35, find the P (2 < x <6)
P(2 < x < 6) = P(x = 3) + P(x = 4) + P(x = 5) =
(Add probabilities from 0 to 5, and subtract the sum of the probabilities from 0 through
2) =
= binomcdf (n,p,5) – binomcdf (n,p,2) =
f) For a binomial experiment with n = 10 and p = 0.73, find the probability that x is
between 4 and 9 inclusive.
P(4 ≤ x ≤ 9) = P(x = 4) + P(x = 5) + ......+ P(x = 9) =
(Add probabilities from 0 to 9, and subtract the sum of the probabilities from 0 through
3) =
= binomcdf (n,p,9) – binomcdf (n,p,3) =
(Page 9 of notes)
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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 (x-max should be 7 here)
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 (x-max should be 11 here)
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
right skewed
If p = 0.5
the shape is
bell shaped
If p > 0.5
the shape is
left skewed
(Page 10 of notes)
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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. Not done during the fall 07
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?
According to the answer given in part (c), it’s not unusual for 15 people to
show up, then overbooking is a real concern.
c) Now use a feature in the calculator to calculate the answer. Indicate what you enter in
the calculator, and the results.
P(not enough seats) = P(x = 15) = binompdf(15, 0.85, 15) = 0.087 (in about 9 out of 100
times)
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.
(Page 11 of notes)
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