Binomial Probability
MATH NSPIRED
TEACHER NOTES
JIM LEFEBER TLC
Objective: Use a simulation to find the experimental probability that a 65% free throw shooter will
make exactly 3 of the next 5 attempts.
Step 1: Create 1 random number.
Enter the formula for random integers from 1 to 100 into the Column A formula cell. =randint(1,100,1)
This number represents a free throw attempt.
The shooter makes 65% of his/her free throws.
A number 1 – 65 will represent a made shot.
66 – 100 will represent a missed shot.
The 11 in cell A1 represents a made shot.
Step 2: Represent 5 free throw attempts.
Copy the formula from the Column A formula cell to the formula cells in column B through E.
Make the columns skinnier by getting the crosshairs over the
line in the Column Name Cell or use
Menu>>Actions>>Resize>>Resize Column Width
In this simulation, the shooter made the first shot (11), missed
the second shot (80), and made the final 3 shots (37, 51, 49).
Step 3: How many did the shooter make out of the 5 shots?
Because the shooter makes 65% of his attempts, the numbers 1 through 65 represent a successful
attempt. Type =countif(a1:e1,?<=66) in cell F1. In the Column Name Cell, label as made.
This means there were 4 times out of 5 that a
number 65 or lower came up. This simulates
the shooter making 4 out of 5 free throws.
Made free throws: 11, 37, 51, 49
Missed free throws: 80
Step 4: Represent 5 trials.
Change the formula cells to =randint(1,100,5). Copy and paste the formula in F1 to F2 through F5 by
holding the shift key and using the down arrow.
Here are 5 simulations.
Each row (1 – 5) represents 5 free throws attempted.
The shooter made 4, 5, 4, 4, and 2 shots (column F).
Binomial Probability
MATH NSPIRED
TEACHER NOTES
JIM LEFEBER TLC
Step 5: Complete the Objective:
Use a simulation to find the experimental probability that a 65% free throw shooter will make exactly 3 of
the next 5 attempts. In Type =countif(f1:f5,3) in cell F6
The spreadsheet counts all the 3’s in the column.
There are no 3’s out of the 5 simulations. The
experimental probability is 0/5 = 0.
Step 6: Create 100 simulations.
Because only 5 simulations were done the answer of zero above is distorted. The more trials performed,
the more reliable the estimate will be.
.
1) Questions based on 100 simulations:
a. Find the experimental probability that a 65% free throw shooter will make exactly 0 of the next 5 attempts.
Answers will vary
b. Find the experimental probability that a 65% free throw shooter will make exactly 1 of the next 5 attempts.
Answers will vary
c. Find the experimental probability that a 65% free throw shooter will make exactly 2 of the next 5 attempts.
Answers will vary
d. Find the experimental probability that a 65% free throw shooter will make exactly 3 of the next 5 attempts.
Answers will vary
e. Find the experimental probability that a 65% free throw shooter will make exactly 4 of the next 5 attempts.
Answers will vary
f. Find the experimental probability that a 65% free throw shooter will make exactly 5 of the next 5 attempts.
Answers will vary
g. Compare your answers to a. and f. Which problem produces a larger probability? Explain why this is.
f. should be higher because it’s more likely a 65% free throw shooter makes all of the shots than no shots.
h. Find the experimental probability that a 65% free throw shooter will make more than 3 of the next 5
attempts.
Answers will vary
i. Find the experimental probability that a 65% free throw shooter will make less than 3 of the next 5 attempts.
Answers will vary
j. Compare your answers to h. and i. Which problem produces a larger probability? Explain why this is.
Binomial Probability
TEACHER NOTES
JIM LEFEBER TLC
MATH NSPIRED
Review Combinations
A combination is an arrangement of objects in which order does not matter (vs. a permutation where order
does matters.
Read "n choose r"
nCr
=
n!
r !(n  r )!
Find the value of each combination
a. 7C5 = 21
b.
5C3 =
10
c.
6C2 =
15
2) Now you can calculate the theoretical probably of the same problem above using the Binomial Probability
formula. In a binomial experiment, the probability of r successes in n trials (0 ≤ r ≤ n) is
P(r)= nCr ∙ pr∙ qn-r, where p is the probability of success and q is the probability of failure.
a. What is the probability that a 65% free throw shooter will make exactly 3 of the next 5 attempts.
P(r)= nCr ∙ pr∙ qn-r, so P(3)=5C3 ∙ .653 ∙ .35(5-3) =
0.336
b. What is the probability that a 65% free throw shooter will make exactly 5 of the next 5 attempts.
P(r)=nCr ∙ pr qn-r , so P(5)=5C5 ∙ .655 ∙ .35(5-5) =
0.116
c. Find the probability that a 65% free throw shooter will make 3 or more of the next 5 attempts.
P(r >3) = P(3) + P(4) + P(5)=
P(4) = 0.312; 0.336+0.312+0.116 = 0.764
d. Find the probability that a 65% free throw shooter will make less than 3 of the next 5 attempts.
P(r <3)= 1 – P(r >3)=
1 – 0.764 = 0.236
Use a simulation to find each experimental probability.
3) An energy drink game advertises a 25% chance of winning with each bottle cap. Find the experimental
probability that a 6-pack will contain at least 3 winners. Use a List and Spreadsheet page to simulate 100
6-packs.
a. In your simulations, how many times are there at least 3 winning caps.
Answers will vary
b. In your simulations, how many times is there exactly 1 winning cap.
Answers will vary
4) Critical Thinking How would you design a simulation to find the probability that a baseball player with a
.285 batting average will get a hit in 5 of his next 10 official at bats (don’t count walks or being hit by pitch)?
Describe what changes would have to be made in this simulation compared to the first two you’ve done.
Need 10 columns
Randint(1,1000,100)
A random number 1 – 285 is a hit. 286 – 1000 is an out
Countif(a1:j1,?<=285)
Binomial Probability
TEACHER NOTES
JIM LEFEBER TLC
MATH NSPIRED
Use P(r)= nCr ∙ pr∙ qn-r to find each theoretical probability.
5) An energy drink game advertises a 25% chance of winning with each bottle cap. Find the probability that a
6-pack will contain the following amounts of winning caps. Round answers to 4 decimal places.
Number of winners
0
1
2
3
4
5
6
P(r)= nCr ∙ pr∙ qn-r
0
(6–0)
6C0 ∙ .25 ∙ .75
Probability
0.1780
0.3560
0.2966
0.1318
0.0330
0.0044
0.0002
Total
1.0000
a. How many winning caps is most likely to occur? 1 winning cap
b. What is the probability of the most likely occurrence? 0.3560
c. What is the most efficient way to answer the question, “What is the probability of getting at least one
winner.” Find the probability of zero winners and subtract from 1.
d. Using the pattern in the table, write an equation that will produce the above table. Use the Entry Line on
a Graph page and Table on the Nspire calculator to check your equation and table. y = 6Cx ∙ .25x∙ .75(6–x)
6) A baseball player with a .285 batting average wants to know the probability of getting 0, 1, 2, 3, … 10 hits in
the next 10 official at bats.
What equation can be typed into the Entry Line on a Graph page to quickly fill in the following table?
y = 10Cx ∙ .285x∙ .715(10–x)
Complete the table using an Nspire calculator
Number of hits
0
1
2
3
4
5
6
7
8
9
10
P(r)=nCr ∙ pr ∙ qn-r
Total
Probability
0.0349
0.1392
0.2497
0.2654
0.1851
0.0885
0.0294
0.0067
0.0010
0.0001
0.000004
1
a. How many hits is most likely to occur? 3 hits
b. What is the probability of the most likely occurrence? 0.2654
c. Explain why this is the most likely occurrence.
3 out of 10 hits is a batting average of .300. The closest to .285 in the table.
Ask: What variables are not being taken into account when doing the batting average probability?
The pitcher he’s facing – could be a bad pitcher, great pitcher, left hander, or right hander injuries, home vs. away, etc.
Extension: Create graphs on the Nspire calculator. See directions on page 5.2