MATH 141-501
Section 8.3 & Section 8.4 Lecture Notes
Bernoulli Trials, Variance, Standard Deviation, and Mean
A binomial experiment is any experiment which has the following properties:
1. The number of trials in the experiment is fixed.
2. There are two outcomes of each experiment. (One is considered a “success”, and the other is a “failure”.
3. The probability of success in each trial is the same.
4. The trials are independent of each other.
Example: A coin is tossed three times, and the number of heads is recorded.
Is this a binomial experiment?
Find the probability of obtaining exactly 0,1,2 or 3 heads.
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Formulas and Examples
Probability of x successes in n repetitions of a binomial experiment: In a binomial experiment where the probability of success is p and the
probability of failure is q = (1 − p), the probability of exactly x successes in n
trials is
C(n, x)px q (n−x) .
Example 1 In a certain county, 3% of registered voters are Libertarians.
If 5 voters are selected at random, what is the probability that exactly 1 is
Libertarian?
What is the probability that all of the randomly selected voters are Libertarian?
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More Examples
Example 2 The graduation rate at a certain university is 80%. If three
freshmen are randomly selected, what is the probability that at least one graduates?
Example 3 How many times must a person toss a fair coin to have the
probability of obtaining at least one head be at least 0.9?
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Expected Value of a Binomial Variable
A coin is tossed 100 times. A toss of heads is considered a success. How
many successes should we expect to see?
Formula: If X is a random variable which represents the number of successes in a binomial experiment with n trials, then the expected value E(X) =
np.
Example: Patients taking an experimental drug have a 40% chance of recovery.. In its first year availability, it is used on 8,000 patients. How many do
we expect will recover?
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Spread
Consider the two lists of numbers below.
List A: 4, 5, 5, 5, 5, 5, 5, 5, 6
List B: 1, 2, 2, 3, 5, 7, 8, 8, 9
Both lists have the number 5 as their mean and median, BUT
List B is much more “spread out” than List A.
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Measures of Spread
Variance and Standard Deviation
Standard deviation and variance are two ways of measuring how much
the entries in a list differ from the mean. They are important numbers in describing a random variable.
As a basic example of the idea, consider
List C: 1, 2, 6
The mean of List C is 3.
To measure the spread of the list, we want to work with the differences of
the entries in the list from the mean.
We can’t just add up the differences (why not?)
Instead, we do this.
x
x−x
Entry Difference from Mean
1
(3 - 1) = 2
2
(3 - 2) = 1
6
(3 - 6) = -3
(x − x)2
Difference from Mean Squared
22 = 4
12 = 1
(−3)2 = 9
Sum of All (x − x)2 = 4 + 1 + 9 = 14
Variance = average of (all differences squared) =
Standard Deviation =
√
14
3
average difference squared =
6
q
14
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Variance and Standard Deviation of Random Variable X
If a random variable X has probability distribution
x
P (X = x)
x1
p1
x2
p2
···
···
xn
pn
and expected value E(X) = µ (“mu”), then the variance of the random
variable X is
Var(X) = p1 (x1 − µ)2 + p2 (x2 − µ)2 + · · · + pn (xn − µ)2 .
The standard deviation σ (“sigma”) is given by
p
σ = Var(X)
p
= p1 (x1 − µ)2 + p2 (x2 − µ)2 + · · · + pn (xn − µ)2
Example(from textbook): Consider the random variables X and Y whose
probability distributions are shown in the following table.
x
1
2
3
4
5
6
7
P (X = x)
.05
.075
.2
.375
.15
.1
.05
y
1
2
3
4
5
6
7
P (Y = y)
.2
.15
.1
.15
.05
.1
.25
Find the mean, variance, and standard deviation of X and Y .
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Measures of Spread for Random Variables in Binomial Experiments
If X is a binomial random variable associated with a binomial experiment
consisting of n trials with probability of success p and probability of failure q,
then the mean, variance, and standard deviation of X are
(Mean)
µ = E(X) = np
(Variance)
Var(X) = npq
√
σX = npq
(Standard Deviation)
Example: A basketball player has made 55% of the free throws in his career. If he shoots 30 free throws in a game, what is the mean and standard
deviation of the number of shots he can be expected to make? (Note: We
assume that each free throw attempt is independent of the others.)
Example: An auto insurance company has determined that the probability of a typical customer getting into an accident during one hour of driving is
approximately 0.0001. The company has 10,000 customers, and the number of
hours driven by a customer is 10 per week.
Find the mean and standard deviation of the number of accidents the company’s customers will be in in a given year.
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