Chapter
4
Probability
Generating Random Numbers
When working with probabilities, there is sometimes a need to generate random numbers.
Computer simulations are a common example of the need for random occurrences within a
structured setting. These numbers are called pseudo-random numbers since they are generated by
a process that is not a random process. However, these pseudo-random numbers behave very
much like random numbers and so are very useful to statisticians. Your calculator can generate
pseudo-random numbers, which will henceforth be called random numbers.
Generating Random Numbers Between 0 and 1
Suppose that you would like to generate a number between 0 and 1. You want the number to be
unpredictable, but you want every number between 0 and 1 to have an equally likely chance of
being generated.
Press
Press
.
three times to reach the PRB menu.
Select 1:rand and press
Press
.
again.
If you continue to press
you will generate a different
random number between zero and one each time you press
.
Note: The values shown in the screenshot will probably not be the same as the values
shown on your calculator screen. These are (pseudo) random numbers!
Generating Random Numbers Between Any Two Values
The TI-84 Plus does not have a built-in function to generate random real numbers that are
equally likely to occur and fall within a specified range of values, but they can be generated by
using the rand function with some additional commands. The following command will generate
random real numbers between 1 and 100.
Press
Press
.
three times to reach the PRB menu.
Select 1:rand and press
.

Press
Press
.
If you continue to press
each time.
, you will generate a different random number between 1 and 100
In general, the command used to generate a real number between values j and k is:
rand * (k – j) + j , where n is the larger number.
For example: rand * (10 – 1) + 1 will generate a real number between 1 and 10.
rand * (900 – 500) + 500 will generate a real number between 500 and 900.
Generating Random Integer Values Between Any Two Numbers
To generate random integer numbers (no decimals) that are equally likely to occur and fall
within a specified range of values, the use the built-in function randInt. The following sequence
will generate 20 random integer values between 1 and 100.
Press
.
Press
three times to reach the PRB menu.
Select 5:randInt( .
.
Press
Press
.
Use
and
to scroll through all twenty numbers. Press
to generate 20 more such
random integers.
Note: The values shown in the screenshot will probably not be the same as the values
shown on your calculator screen. These are (pseudo) random numbers!
2
In general, the command used to generate n integer numbers between values j and k is:
randInt(j, k, n)
If you do not enter a value for n, then the calculator will use 1 as the default value for n.
For example: randInt(10, 50, 15) will generate 15 random integers between 10 and 50.
randInt(10, 50) will generate 1 random integer between 10 and 50.
Generating Random Integers without Repetitions
In some applications, random integers need to be generated without repeating any values. For
example, suppose you wish to draw ten numbered chips from a bag, but you do not replace any
chips once they are drawn. The TI-84 Plus calculator can simulate this process.
Press
.
Press
three times to reach the PRB menu.
Select 8:randIntNoRep(.
.
Press
Press
.
The integers from 1 to10 (inclusive) will be randomly arranged as if they were the numbers on
chips being drawn from a bag without replacement. Use
numbers. Press
and
to scroll through all ten
to generate a different sequence of random draws.
In general, the command used to randomly generate values between j and k without repetition is:
randIntNoRep(j, k)
For example: randIntNoRep(10, 20) will randomly generate integers between 10 and 20
without repetition.
Store Random Numbers in a List
The random numbers generated can be stored in a list to be
used with other statistical procedures.
The command in the screen
shot on the right will generate
15 randomly-generated integer
values between 10 and 50 and
store them in List L1.
3
Calculating Factorials, Combinations, and Permutations
Factorials
A common function needed to compute probabilities is the
factorial function. The notation for the factorial of n is n! The
“!” function is found by pressing
menu.
and selecting the PRB
To find the number of ways six people could be arranged in six
different chairs, you would calculate six factorial (6!).
Press
.
Press
.
Press
to reach the PRB menu.
Select 4: !.
Press
.
Calculate 10! and 0!.
Combinations
The combination formula can also be used to compute probabilities. The notation for the number
of combinations is nCr, where n is the total number of elements, and r is the number being
selected. Combinations are used when selecting a few elements from a larger number of distinct
elements.
Example: Ice Cream
An ice cream parlor offers 6 flavors of ice cream. Jason would like to purchase 2 flavors of ice
cream. In how many ways can Jason choose 2 flavors out of the
6 flavors?
In order to find the number of ways of choosing two flavors out
of six, we would need to calculate 6C2.
Press
Press
.
.
Press
to reach the PRB menu.
Select 3: nCr.
Press
press
.
4
There are 15 different combinations of two flavors of ice cream.
Calculate 6C3 and 8C3.
Permutations
The permutation formula can be used to compute probabilities. The notation for the number of
permutations is nPr, where n is the total number of elements, and r is the number being selected.
Permutations are used when trying to find all possible arrangements of elements taken from a
larger group. Arrangements involve putting the elements in a particular order.
If Jason’s story changes as below, then permutations apply
rather than combinations.
An ice cream parlor offers 6 flavors of ice cream. Jason would
like to purchase 2 flavors of ice cream and is concerned about
which flavor is on the top and which flavor is on the bottom (i.e.
Jason is concerned about the arrangement of the flavors). In
how many ways can Jason arrange 2 flavors out of the 6
flavors?
In order to find the number of arrangements of choosing two flavors out of six, we would need to
calculate 6P2.
Press
Press
.
.
Press
to reach the PRB menu.
Select 2: nPr.
Press
press
.
There are 30 different arrangements of two flavors of ice cream.
Calculate 6P3 and 8P3.
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