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Lesson 4.3 The Binomial Probability
Distribution
Bernoulli Trial
Suppose we toss a fair coin repeatedly. Each toss
is called a trial. For each trial, the probability
of getting heads is 1/2. This probability does not
change from trial to trial. These types of trials
where the probability does not change are
independent. When there are only two outcomes,
they are called Bernoulli Trials named after James
Bernoulli who researched them at the end of the
17th century.
In a Bernoulli Trial, there are only two outcomes
- "success" and "failure." The random variable
defines a "success."
The probability of a "success" is p and the
probability of a "failure" is q.
p + q = 1.
Characteristics of the
Binomial
A binomial experiment consists of counting the
number of successes in one or more Bernoulli
Trials.
The random variable X is equal to the number of
successes.
- n = the number of trials
- p = the probability of a
success on any trial
Each trial is independent.
Notation and Formulae
The random variable X counts the number of
successes. X takes on the values0, 1, 2, 3, 4,
..., n.
X follows a binomial distribution with parameters
n and p. We write this as
X ~ B(n, p)
Shortcut formulae:
mean: μ = np
Variance: σ 2 = npq
Standard deviation: σ = square root of npq.
Binomial Problems Using
TI-83 or TI-84 calculators
Example: John comes to class totally unprepared
for a 20 question true-false quiz, so he guesses
randomly.
 
Let X = the number of questions John guesses
correctly (guessing correctly is a "success.")
X takes on the values 0, 1, 2, 3, ..., 20.
n = 20 questions (number of trials)
p = 1/2 (This is a true-false quiz. John is
totally unprepared, so he has a 50% chance of
guessing correctly on each question.)
X ~ B(20, 1/2)
mean: μ = np = (20)(1/2) = 10 questions
Variance: σ 2 = npq =
(20)(1/2)(1/2) =5
Standard deviation: σ = square root
of 5 = 2.24 questions
The following are some binomial probability
problems:
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1. Find the probability that John
guesses 7 questions correctly. (Find P(X
= 7).)
P(X = 7) = 0.0739
This calculation was done using the
TI-83 or TI-84 calculator function.
2nd DISTR 0:binompdf( 20, 1/2, 7)
2. Find the probability that John
guesses more than 7 questions correctly.
(Find P(X > 7)).
P(X > 7) = 0.8684
This calculation was done by entering
"1 - " on the home screen of TI-83 or
TI-84 calculator and then entering the
calculator function
2nd DISTR A:binomcdf (20, 1/2, 7)
3. Find the probability that John
guesses 6 or 7 questions correctly.
(Find P(X = 6) + P(X = 7)).
P(X = 6) + P(X = 7) = 0.0370 +
0.0739 = 0.1109
This calculation was done using TI-83
or TI-84 calculator functions
2nd DISTR 0:binompdf( 20, 1/2, 6) + 2nd
DISTR 0:binompdf( 20, 1/2, 7)
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Example
A problem
involving statistics students and their homework
is a typical binomial probability problem that is
solved by using TI-83 or TI-84 calculators. Close
the window when you are finished viewing the
example. You will return here.
Please continue to the next section of this lesson.
Up » 4.1
Discrete Probability »
4.2 Expected Value
» 4.3 Binomial
Probability » 4.4 Poisson Probability
Lesson 1
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