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Introductory Statistics
Lesson 4.2 A
Objective:
SSBAT determine if a probability experiment is a binomial
experiment.
SSBAT how to find binomial probabilities using the
binomial probability formula.
Standard: S2.5B
Binomial Experiment
A Probability Experiment that satisfies the following:
1. The experiment is repeated for a fixed number of trials,
where each trial is independent of the other trials.
2. There are only 2 possible outcomes for each trial. The
outcomes can be classified as a success (S) or a failure (F)
3. The probability of a success P(S) is the same for each trial
4. The random variable x counts the number of successful
trials
Notations for Binomial Experiments
n =
Number of times a trial is Repeated
p =
Probability of Success P(S)
q =
Probability of Failure P(F)
x =
Represents a count of the number of
successes in n trials: x = 0, 1, 2, 3, …, n
Examples: Find the following for each
a) Determine if it is a Binomial Experiment
b) If it is, specify the n, p, and q values, and list the possible values of x
1. You select a card from a standard deck, note whether or not
it is a club, then replace the card. You repeat this
experiment 5 times.
a) Yes – it satisfies all 4 conditions
b) n = 5
p = P(S) =
13
52
= 0.25
q = P(F) = 1 – 0.25 = 0.75
x
= 0, 1, 2, 3, 4, 5
2. A certain surgical procedure has an 85% chance of
success. A doctor performs the procedure on eight patients.
The random variable, x, represents the number of
successful surgeries.
a) Yes – it satisfies all 4 conditions
b) n = 8
p = P(S) = 0.85
q = P(F) = 0.15
x
= 0, 1, 2, 3, 4, 5, 6, 7, 8
3. A jar contains five red marbles, nine blue marbles, and six
green marbles. You randomly select three marbles from the
jar, without replacement. The random variable represents
the number of red marbles.
a) Not a Binomial Experiment
 The marble is not replaced so each trial
does not have the same probability of
success. The probability of success
changes after each trial.
4. You take a multiple choice quiz that consists of 10 questions.
Each question has four possible answers, only one of which is
correct. To complete the quiz, you randomly guess the answer to
each question. The random variable represents the number of
correct answers.
a) Yes – it satisfies all 4 conditions
b) n = 10
p = P(S) = ¼ = 0.25
q = P(F) = 1 – 0.25 = 0.75
x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
5. Tom makes 79% of his free-throw attempts. He attempts
12 free-throws at practice. The random variable
represents the number of successful free-throws.
a) Yes – it satisfies all 4 conditions
b) n = 12
p = P(S) = 0.79
q = P(F) = 1 – 0.79 = 0.21
x = 0, 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10, 11, 12
 There are several ways to find the probability of x
successes in n trials. One is using a tree diagram and
the multiplication rule, like we did in Chapter 3.
 Another way is to use the Binomial Probability
Formula
Binomial Probability Formula
 The probability of exactly x successes in n trials is:
P(x) =
nCx
px
qn-x
=
𝑛!
𝑛−𝑥 !𝑥!
𝑝 𝑥 𝑞 𝑛−𝑥
Examples.
1. Microfracture knee surgery has a 75% chance of success on
patients with degenerative knees. The surgery is performed on
3 patients. Find the probability of the surgery being successful
on exactly 2 patients. (source: Illinois Orthopaedic and Sports Medicine Centers)
P(x) =
nCx
px qn-x
 Identify the variables
n = 3, p = .75, q = .25, x = 2
 P(2 successes) = 3C2 (.75)2 (.25)1
P(2 successes) ≈ 0.422
2. A card is selected from a standard deck of cards and
replaced. This experiment is repeated a total of 5 times.
Find the probability of selecting exactly 3 clubs.
P(x) =
 n = 5,
p=
13
52
nCx
= 0.25,
px qn-x
q = 1 – 0.25 = 0.75,
P(3 clubs) = 5C3 (0.25)3 (0.75)2
P(3 clubs) ≈ 0.088
x=3
3. A multiple-choice quiz consists of 8 questions. Each
question has 5 possible choices, with only 1 correct answer.
If you forgot to study and randomly guess on each question,
what is the probability you get exactly 5 questions correct?
P(x) =
 n = 8,
p=
1
5
nCx
px qn-x
= 0.2,
q = 1 – 0.2 = 0.8,
P(5 successes) = 8C5 (.2)5(.8)3
P(5 successes) ≈ 0.009
x=5
Homework
Worksheet 4.2 A