Binomial Distribution Formula

The Binomial Distribution:
Objectives
• To Introduce the notion of a ‘Bernoulli Trial’
• To introduce the Binomial Probability Distribution
as the situation when a finite number of Bernoulli
Trials is conducted
• To recognise problems suitable for Binomial
Probability modeling and calculate Binomial
probabilities using a formula
• To understand and apply the Binomial Distribution
in hypothesis testing: Binomial Test

1
The Binomial Distribution. Max Chipulu, University of Southampton 2009

Try this

Have you got a coin?
Toss it six times in a roll, each time counting
the number of times the result, ‘heads’ is
heads’
observed.

2

Observed Probability

Before a coin is tossed six times in a roll,
what is the probability that in total there will
be two ‘heads’ out of six?
heads’

3

A BERNOULLI TRIAL

Tossing a coin is Bernoulli trial. A Bernoulli trial is a
random experiment that has only two, mutually
exclusive outcomes.

Thus, when a coin is tossed, the two possible outcomes
are revealing a ‘heads’ or revealing a ‘tails’. We want to
heads’ tails’
see how many ‘heads’ are revealed. So revealing a
heads’
‘heads’ is a ‘success’. Revealing a ‘tails’ is a ‘failure’.
heads’ success’ tails’ failure’

4

Properties of a BERNOULLI TRIAL

Since you’re tossing the same coin in all six
you’
Bernouli trials, the probability of a ‘heads’ or a
heads’
success, p, is the same for each repeat of the
Bernoulli trial. stationarity
assumption
But the coin has no memory: Bernoulli trials are
independent.
independence
assumption
5

Multiple Answer Question: Which of the
following are Bernoulli Trials

• A: The Lotto draw
• B: The experiment: randomly select a company
from all public limited companies in the UK. If the
company is in Liquidation, the experiment is
successful. If the company is in the Biomedical
industry, the experiment is a failure.
• C: Randomly selecting balls from a population
containing white balls and black balls

6

Multiple Answer Question: Which of the
following are repeats of Bernoulli Trials

• D: The game of joker’s challenge is played with
joker’
the full deck of cards: The player is shown a card,
and s/he calls whether the next card will be lower
or higher, if the call is correct the game is
repeated, and so on
• E: A box contains 20 packs of chocolate, 8 of
which are milk. Selecting a chocolate, checking if
its milk, eating eat it, really enjoying it and then
repeating the exercise.
• F: The gender of a randomly selected CEO

7

Binomial Probability

Before a coin is tossed six times in a roll, what
is the probability that in total there will be two
‘heads’ out of six?
heads’

Method 1: The painstaking, torturous way:

List all possible results that the arise from the
six coin tosses. Count the total. Count how
many times 2 'heads‘ are revealed. Calculate
'heads‘
the probability.

8

Method 2: The easy Way. Use a formula.

Somebody has already done all this before. There is a
formula. It tells us how to obtain i things out of n.

i n!
It is : C n =
i! ( n − i )!
Where n! or ‘n factorial’ = n(n-1)(n-2)(n-3)…3*2*1;

Thus, the total number of times 2 heads can be
revealed by the six coin tosse s is :
6 * 5 * 4 * 3 * 2 *1
C 62 = = 15
( 2 * 1 * ( 4 * 3 * 2 * 1))
9

Question: What is the probability that in total 2
coins out of six tossed will be heads?

First, the Addition Rule

We know there are 15 ways of selecting 2
successes out of six trials. So the probability
we want is one these 15 combinations:

P(x = 2) = P(HHTTTT or HTHTTT or HTTHTT
or HTTTHT or HTTTTH or THHTTT etc)

10

Mutual Exclusivity
But we can only have one combination revealing two
people at any given time:

For example its either HTHTTT or HTTHTT, not
HTTHTT,
both

They are mutually exclusive. So the Addition rule
simplifies to:

P(i = 2) = P(HHTTTT) +P(HTHTTT) + P(HTTHTT)
+ P(HTTTHT) + P(HTTTTH) + ….

11

Second, the Multiplication Rule
What is the probability that a sequence of results such
as HHTTTT will be revealed?

P(HHTTTT) = P(H)*P(HTTTT|H)

We want to know: What is the probability that the
sequence HTTTT will be revealed successively, if the
first result is heads?

But successive trials are independent: Whether or not the
first result is heads does not affect the results of the next
five trials.
So we simplify: P(HHTTTT) = P(H)*P(HTTTT|H) =
P(H)*P(HTTTT) 12

The Multiplication Rule, Cont’d
Cont’

In the same way, we can expand P(HTTTT) =
P(H)*P(TTTT)

And so on, and so forth, so that in the end:

P(HHTTTT) = P(H)*P(H)*P(T)P*(T)*P(T)*P(T)
P(HTTTTT) = P(H)2P(T)4 = 0.52*0.54 = 0.015625

13

The Multiplication Rule, Cont’d
Cont’

A little thought should show that all the other
combinations two heads out of six trials have the same
probability:

P(HHTTTT) =P(HTHTTT) = P(HTTHTT)
=P(TTTHTT) =P(HTTTTH) =P(HTTTTH), etc..

Hence P(2 heads out of six trials) = 15 * 0.52*0.54 =
0.2344

14

2 4
Binomial Distribution
successes
failures
We obtained these values as follows:
P(2 Sufferers) = 15 * 0.52*0.54

C 62 = 15 P(failure)
=1 - p =
P(success) 0.5
= p = 0.5

15

We can generalise this expression so that we are
able to calculate the probability for any number of
successes i in n Bernoulli trials, for which the
probability of success of each is p:

P(x = i) = Cn p i (1 − p)n −i
i

Number of n – i failures.
combinations of i i successes. P(failure) =1 - p
out of n P(success) = p

This is the binomial distribution 16

We can derive the expected value and the
variance for this distribution.
They are:

µ = np σ 2 = np(1 − p)

17

The Binomial Distribution Example: Mortgage Default
Rates

• It is estimated that 1.5% of home owners (with a
mortgage) will default on their mortgage. Suppose a
random sample of 12 home owners is taken. What is the
probability that
• (a) none of them will default
• (b) at least one of them will default
• Is the probability that all of them will default the same as
none of them will default?
• Suggested calculation steps
• 1. This a Binomial Distribution- why?
• 2. What is the trial, what is the success, what is the
probability of a success?
• 3. Calculate probabilities using the formula
18

Solution

19

Solution Continued

20

Solution Continued

21

Solution Continued

22

Experiment: All Colas Taste
Same

Need Ten Volunteers

Ten people that love colas

23

Example: Cola Flavours in a Supermarket (1997/98 Exam)

A product manager is in charge of a cola in a supermarket. His superior,
the managing director, and the publicity department of the company,
claim that this cola, Supercola, has a very distinctive flavour. The
product manager is of the opinion that, without seeing the label, people
are unable to discriminate between colas. In order to test his theory,
the product manager conducts an experiment. He asks ten individuals to
assess if the liquid in a cup is Supercola or one of the well-known
brands: Pocacola and Colaloca. The results are as follows:

24

Example: Cola Flavours in a Supermarket (1997/98 Exam)

1 Colaloca Supercola
2 Supercola Colaloca
3 Colaloca Pocacola
4 Pocacola Pocacola
5 Supercola Colaloca
6 Colaloca Supercola
7 Pocacola Pocacola
8 Pocacola Supercola
9 Supercola Supercola
10 Colaloca Pocacola

25

(a) What can the product manager conclude from the above
experiment? Identify a probability model and use it to arrive
at a conclusion.

Explain your reasoning and give any relevant equations.

(b) The managing director is of the opinion that ten trials is a very
small number on which to base a marketing strategy, and insists
that a larger experiment of 300 individuals should be
conducted. You are requested to calculate the range of values
within which it can be concluded that people cannot discriminate
between colas. Explain your answer and give any relevant
equations.

26

Binomial Test Example: Are Greek SMEs Risk Averse?
As part of her dissertation into the risk appetite of Greek SMEs (Small to Medium
Enterprises) Margarita Georgousopoulou (2009 Risk Management
Dissertation) asked respondents to rate their response on a Likert scale with
categories 1 to 5, where by 1= ‘Very Risk Seeking’, 2 = ‘Risk Seeking’, 3 =
‘Risk Neutral’, 4 = ‘Risk Averse’ and 5 = ‘Very Risk Averse’.
She tested several variables on this scale.
For the ‘Machinery Investment’ Variable, the results were as in the table below:

Can it be concluded that Greek SMEs are risk averse vis-à-vis ‘machinery
investment’?
27

Greek SMEs Solution
• Define two events, a success and a failure, such
that:
• Event = ‘Success’ if response value is greater than
3, i.e. SME is Risk Averse
• Event = ‘Failure’ if response value is 3 or less, i.e.
SME is NOT risk averse
• From the table, number of successes, i = 24.
Number of trials = Sample size = 54.
• Why is the binomial distribution a reasonable model
for this situation?
• Binomial Test question: is 24 successes out of 54
trials statistically significant?
28

Greek SMEs: Binomial Test in SPSS 17
• Load the data file from blackboard called ‘Risk
Appetite.xls’ into SPSS.
• Select the worksheet called ‘raw data’
• From the ‘Analyze’ menu, select ‘nonparametric tests’
• Select ‘Binomial’
• Enter the variable ‘machinery investment’ into the
‘test variable box’
• In ‘Define Dichotomy’, select ‘cut point’. Enter ‘3’ in
the cut point box. This tells SPSS that the values equal to
or less than 3 are classed as one category, while values
above are in the other category (hence the ‘dichotomy’,
which refers to binary categories)

29

Greek SMEs: Binomial Test in SPSS 17 Cont’d

• In the box labeled ‘test proportion’, enter
‘0.4’:This will tell SPSS that under the null
hypothesis, the expected proportion of values above
3 is 40% (this is because 2 of the five categories, i.e.
4 and 5 are higher than 3). Therefore the Binomial
test is to see if the observed proportion (of 24/54)
is significantly higher than 0.4
• Press ‘ok’ to run the model.

30

Greek SMEs: Binomial Test in SPSS 17 Cont’d
Or better still, press ‘paste’: This will paste the program syntax
into a new syntax or program window. Whatever you do in
SPSS, it is based on some program syntax. If you just use the
menus, you cannot normally see the syntax. Yet, it is more
efficient to run the syntax rather than use the menus, especially
if you wish to repeat the analysis, since all you have to do is run
the syntax again. Even better you can run tests on other
variables simply by changing the variable name in the syntax.
You could also change the test characteristics, such as the cut
point and the test proportion. Furthermore, via the brilliant
‘copy’ and ‘paste’ keyboard controls, you can run all the
analysis in one go, as one program. TRY THIS: you will feel
better!!
• To run the model, press the ‘run’ button in the syntax
window, which looks like the ‘play’ button on a CD player.
31

Greek SMEs: SPSS Binomial Test Result

• The significance of the binomial test is given under the column
‘Asymp. Sig.’ This is a one-tailed test, since we’re only testing
whether the observed proportion is greater than 0.4, so that our test
is only one side of 0.4. The test would be two-tailed if we were
testing the hypothesis that the proportion is exactly equal to 0.4
• The value of significance is 0.015. What can we conclude from this?

32

Greek SMEs: Binomial Test in SPSS 17 Exercise

Please try this at home: Following the steps above,
test whether or not the Greek SMEs are risk
averse in the following variables.
(i) ‘new product investment’;
(ii) ‘new employee investment’; and
(iii) ‘new market investment’

33

Binomial Distribution Formula

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Binomial Distribution Formula