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381
Discrete Probability Distributions
(The Binomial Distribution)
QSCI 381 – Lecture 13
(Larson and Farber, Sect 4.2)
Binomial Experiments-I
381

1.
2.
3.
4.
Binomial experiments are those for which the
outcome from each trial is one of only two options
(“success” or “failure”). The properties of a
binomial experiment are:
The experiment is repeated for a fixed number of
trials, where each trial is independent of all the
others.
There are only two possible outcomes for each
trial. The outcomes can be classified as a success
(S) or a failure (F).
The probability of success is the same for all trials.
The random variable X counts the number of
successful trials out of n trials.
Binomial Experiments-II
381

Why is the following a binomial
experiment?



We randomly sample 500 fish from the
population.
We record whether each animal is mature
or immature.
The random variable X is the number of
mature animals.
Binomial Experiments-II
(Notation)
381
1. The experiment is repeated for a fixed number of trials
2. There are only two possible outcomes (S and F)
3. The probability of success P(S) is the same for each trial
4. The random variable x counts the number of successful trials
New
terms
n
The number of times a trial is repeated
p  P ( S ) The probability of a success in a single trial
q  P ( F ) The probability of a failure in a single trial, q  1- p
x
The number of successes on n trials (x=0,1,..,n)
Binomial Probabilities-I
381

In a binomial experiment, the probability of
exactly x successes in n trials is:
P( x)  n Cx p x q n  x 


n!
p x q n x
(n  x)! x !
The binomial probability therefore involves the
probability of x successes and n -x failures
multiplied by the number of ways choosing x
successes out of n trials.
n and p are known as parameters. Much of
statistics involves using data to estimate the
values for unknown parameters.
Binomial Probabilities-II
381

Notation:
X ~ B(n; p)
We read this as “The random variable X is
distributed binomially with parameters n and
p”.
Mean :   n p
Variance :  2  npq
Binomial Probabilities-III
381

By listing all possible values of x with
the corresponding probability of
each, you can construct a
.
Binomial Probabilities-IV
381

There is a probability of 0.2 that a fish in a given population has a
particular disease. Assuming that 5 fish are sampled, construct the
binomial probability distribution for the experiment.
P(0)  5 C0 (0.2)0 (0.8)5  0.3277
P(1)  5 C1 (0.2)1 (0.8) 4  0.4096
P(2)  5 C2 (0.2) 2 (0.8)3  0.2048
P(3)  5 C3 (0.2)3 (0.8) 2  0.0512
P(4)  5 C4 (0.2) 4 (0.8)1  0.0064
P(5)  5 C5 (0.2)5 (0.8)0  0.0003

What does this tell you about a sample size of 5 in this case?
The Binomial Distribution
381
0.6
0.6
n=6; p=0.5
0.5
0.5
Probability
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
1
0
2
1
3
2
4
3
5
4
6
5
7
6
10
12
23
34
5
4
65
7
6
54
65
76
0.6
0.5
n=6; p=0.9
0.4
Probability
Probability
0.4
n=6; p=0.7
0.3
0.2
0.1
0.0
10
2
1
32
4
3
Examples of the Binomial
Distribution-I
381

We examine 12 animals for the presence of
a disease (p=0.1). What is the probability
that:
1.
2.
3.

We find exactly 2 animals with the disease?
We find no animals with the disease?
We find 2 or more animals with the disease?
How many animals do we need to examine
to be 99% sure that at least one has the
disease?
Examples of the Binomial
Distribution-II
381
0.40
0.35
0.30
Probability
0.25
0.20
0.15
0.10
0.05
0.00
01
12
32
43
45
65
76
78
89
9
10
10
11
11
12
12
13
Hint: I used the EXCEL function “COMBIN(N,x)”
Examples of the Binomial
Distribution-III
381
1.
2.
3.

12  2 10
  0.1 0.9  0.2301
2
12  0 12
  0.1 0.9  0.2824
0
P[X=2]=
P[X=0]=
P[X2]=1-P[X=0]-P[X=1]=0.3410
We want to find n such that
1-P[X=0] < 0.01.
This leads to n=40.
The Negative Binomial
Distribution-I
381



We have an experiment with two
outcomes: success (with probability p)
and failure (with probability q =1-p).
Let r be a fixed number of successes,
and the random variable X be the
number of failures before we have r
successes.
The probability distribution for X is:
 x  r  1 r x
P( x)  
p q
 x 
The Negative Binomial
Distribution-II
381

The product term p r q x is multiplied by x  r 1 Cx
and not xr Cx because the final success is
always the result of the last “trial” so we
“know” when the last success occurs.
The Geometric Distribution-I
381


This is a special case of the negative
binomial distribution for which r =1 (i.e.
the probability of the number of failures
until one success is recorded).
What then is the probability of finding the
first diseased animal after finding five that
are not diseased?
P[ X  5]  0.1x 0.95  0.059
381
The Geometric Distribution-II

1.
2.
3.
The Geometric distribution can be
developed from the assumptions that:
A trial is repeated until a success
occurs.
The repeated trials are independent of
each other.
The probability of success p is
constant for each trial.
381
The Geometric Distribution-III

What is the probability that the first
diseased fish is not one of the first four
examined?

This is equivalent to saying that the number
of failures is NOT 0, 1, 2, or 3, i.e.:
1  P[ X  0]  P[ X  1]  P[ X  2]  P[ X  3]
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