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MA 320-001: Introductory Probability
David Murrugarra
Department of Mathematics,University of Kentucky
http://www.math.uky.edu/~dmu228/ma320/
Spring 2017
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 1 / 12
Bernoulli Experiment
A Bernoulli experiment is a random experiment, the outcome ofwhich can be classified in one of two mutually exclusive andexhaustive ways–say, success or failure.
Let X be a random variable associated with a Bernoulli trial by definingit as follows:
X (success) = 1 and X (failure) = 0.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 2 / 12
Bernoulli Experiment
A Bernoulli experiment is a random experiment, the outcome ofwhich can be classified in one of two mutually exclusive andexhaustive ways–say, success or failure.
Let X be a random variable associated with a Bernoulli trial by definingit as follows:
X (success) = 1 and X (failure) = 0.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 2 / 12
Bernoulli Trials
A sequence of Bernoulli trials occur when a Bernoulli experiment isperformed several independent times so that the probability of successremains the same from trial to trial.
Let p denote the probability of success in each trial and q = 1− p theprobability of failure.
ExampleConsider the experiment of flipping a fair coin five independent times.
The probability of heads on any one toss is 1/2.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 3 / 12
Bernoulli Trials
A sequence of Bernoulli trials occur when a Bernoulli experiment isperformed several independent times so that the probability of successremains the same from trial to trial.
Let p denote the probability of success in each trial and q = 1− p theprobability of failure.
ExampleConsider the experiment of flipping a fair coin five independent times.
The probability of heads on any one toss is 1/2.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 3 / 12
Bernoulli Trials
A sequence of Bernoulli trials occur when a Bernoulli experiment isperformed several independent times so that the probability of successremains the same from trial to trial.
Let p denote the probability of success in each trial and q = 1− p theprobability of failure.
ExampleConsider the experiment of flipping a fair coin five independent times.
The probability of heads on any one toss is 1/2.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 3 / 12
Binomial Probabilities
We shall be interested in the probability that in n Bernoulli trials thereare exactly j successes.
DefinitionGiven an n Bernoulli trial with probability p of success, the probabilityof exactly j successes is denoted by b(n,p, j).
ExampleCalculate b(3,p,2).
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 4 / 12
Binomial Probabilities
We shall be interested in the probability that in n Bernoulli trials thereare exactly j successes.
DefinitionGiven an n Bernoulli trial with probability p of success, the probabilityof exactly j successes is denoted by b(n,p, j).
ExampleCalculate b(3,p,2).
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 4 / 12
Binomial Probabilities
We shall be interested in the probability that in n Bernoulli trials thereare exactly j successes.
DefinitionGiven an n Bernoulli trial with probability p of success, the probabilityof exactly j successes is denoted by b(n,p, j).
ExampleCalculate b(3,p,2).
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 4 / 12
Binomial Probabilities
Let the random variable X equal the number of observed successes inn Bernoulli trials, then the possible values of X are 0,1,2, . . . ,n.
If x successes occur, where x = 0,1,2, . . . ,n, then n − x failuresoccur. The number of ways of selecting x positions for the x successesin the n trial is (
nx
)=
n!x!(n − x)!
.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 5 / 12
Binomial Probabilities
Let the random variable X equal the number of observed successes inn Bernoulli trials, then the possible values of X are 0,1,2, . . . ,n.
If x successes occur, where x = 0,1,2, . . . ,n, then n − x failuresoccur.
The number of ways of selecting x positions for the x successesin the n trial is (
nx
)=
n!x!(n − x)!
.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 5 / 12
Binomial Probabilities
Let the random variable X equal the number of observed successes inn Bernoulli trials, then the possible values of X are 0,1,2, . . . ,n.
If x successes occur, where x = 0,1,2, . . . ,n, then n − x failuresoccur. The number of ways of selecting x positions for the x successesin the n trial is (
nx
)=
n!x!(n − x)!
.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 5 / 12
Binomial Distribution
f (x) =(
nx
)px(1− p)n−x , x = 0,1,2, . . . ,n.
These probabilities are called binomial probabilities, and the randomvariable X is said to have a binomial distribution.
A binomial distribution will be denoted by the symbol b(n,p), and wethat the distribution of X is b(n,p).The constants n and p are called theparameters of the binomial distribution.
∑x∈S
f (x) = 1
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 6 / 12
Binomial Distribution
f (x) =(
nx
)px(1− p)n−x , x = 0,1,2, . . . ,n.
These probabilities are called binomial probabilities, and the randomvariable X is said to have a binomial distribution.
A binomial distribution will be denoted by the symbol b(n,p), and wethat the distribution of X is b(n,p).The constants n and p are called theparameters of the binomial distribution.
∑x∈S
f (x) = 1
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 6 / 12
Binomial Distribution
f (x) =(
nx
)px(1− p)n−x , x = 0,1,2, . . . ,n.
These probabilities are called binomial probabilities, and the randomvariable X is said to have a binomial distribution.
A binomial distribution will be denoted by the symbol b(n,p), and wethat the distribution of X is b(n,p).The constants n and p are called theparameters of the binomial distribution.
∑x∈S
f (x) = 1
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 6 / 12
Binomial Distribution
Figure: Binomial density function.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 7 / 12
Binomial Distribution
f (x) =(
nx
)px(1− p)n−x , x = 0,1,2, . . . ,n.
µ = E(X ) = np.
σ2 = npq
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 8 / 12
Binomial Distribution
f (x) =(
nx
)px(1− p)n−x , x = 0,1,2, . . . ,n.
µ = E(X ) = np.
σ2 = npq
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 8 / 12
Cumulative Distribution Function
The cumulative distribution function or, more simply, thedistribution function of the random variable X is
F (x) = P(X ≤ x), −∞ < x <∞,
For the binomial distribution the distribution function is defined by
F (x) = P(X ≤ x) =bxc∑y=0
(ny
)py (1− p)n−y
where bxc is the floor or greatest integer less than or equal to x .
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 9 / 12
Binomial Distribution
Figure: Binomial distribution cdf.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 10 / 12
Binomial Distribution
ExampleConsider the experiment of flipping a fair coin six independent times.What is the probability that exactly three heads turn up?
b(6,0.5,3) =(
63
)(12
)3(12
)3
= 0.3125.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 11 / 12
Binomial Distribution
ExampleConsider the experiment of flipping a fair coin six independent times.What is the probability that exactly three heads turn up?
b(6,0.5,3) =(
63
)(12
)3(12
)3
= 0.3125.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 11 / 12
Binomial Distribution
ExampleConsider the experiment of rolling a fair die four independent times.What is the probability that exactly one six?
We this experiment as a 4 Bernoulli trials withsuccess: "rolling a 6" andfailure: "rolling some number other than 6".
Then p = 1/6.
b(4,1/6,1) =(
41
)(16
)1(56
)3
= 0.386.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 12 / 12
Binomial Distribution
ExampleConsider the experiment of rolling a fair die four independent times.What is the probability that exactly one six?
We this experiment as a 4 Bernoulli trials withsuccess: "rolling a 6" andfailure: "rolling some number other than 6".
Then p = 1/6.
b(4,1/6,1) =(
41
)(16
)1(56
)3
= 0.386.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 12 / 12
Binomial Distribution
ExampleConsider the experiment of rolling a fair die four independent times.What is the probability that exactly one six?
We this experiment as a 4 Bernoulli trials withsuccess: "rolling a 6" andfailure: "rolling some number other than 6".
Then p = 1/6.
b(4,1/6,1) =(
41
)(16
)1(56
)3
= 0.386.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 12 / 12
Binomial Distribution
ExampleConsider the experiment of rolling a fair die four independent times.What is the probability that exactly one six?
We this experiment as a 4 Bernoulli trials withsuccess: "rolling a 6" andfailure: "rolling some number other than 6".
Then p = 1/6.
b(4,1/6,1) =(
41
)(16
)1(56
)3
= 0.386.
David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 12 / 12
