This document provides an overview of the binomial probability distribution, including key terminology like random experiments, outcomes, sample space, and discrete vs. continuous random variables. It defines a binomial experiment as having n repeated trials with two possible outcomes (success/failure), where the probability of success p is constant for each trial. The number of successes is a binomial random variable with a binomial probability distribution. Several examples are given to illustrate calculating probabilities of outcomes for binomial experiments involving dice rolls, patient recoveries, telephone call successes, ratios of children's sexes, and metal piston rejects. The mean, variance, and standard deviation of the binomial distribution are also defined in terms of n, p, and q.

1. BINOMIAL
PROBABILITY
DISTRIBUTION

2. BASICS & TERMINOLOGY:
 Outcome:- The end result of an experiment.
 Random experiment:- Experiments whose outcomes are not
predictable.
 Random Event:- A random event is an outcome or set of
outcomes of a random experiment that share a common attribute.
 Sample space:- The sample space is an exhaustive list of all
the possible outcomes of an experiment, which is usually denoted
by S.

3.  Random Variables.
 Discrete Random Variable .
 Continuous Random Variable.
 Bernoulli Trial:
A trial having only two possible outcomes is called
Bernoulli trials.
example:
 success or failure
 head or tail

4.  Binomial Distribution:-
The Binomial Distribution describes discrete , not
continuous, data, resulting from an experiment known
as Bernoulli process.

5. A binomial experiment is one that
possesses the following properties:
 The experiment consists of n repeated trials;
 Each trial results in an outcome that may be classified as a success
or a failure (hence the name, binomial);
 The probability of a success, denoted by p, remains constant from trial
to trial and repeated trials are independent.
 The number of successes X in n trials of a binomial experiment is
called a binomial random variable.
 The probability distribution of the random variable X is called a
binomial distribution, and is given by the formula:
P(X) = Cn
xpxqn−x

6. EXAMPLE 1
A die is tossed 3 times. What is the probability of
(a) No fives turning up?
(b) 1 five?
(c) 3 fives?

8. EXAMPLE 2:
Hospital records show that of patients suffering from
a certain disease, 75% die of it. What is the
probability that of 6 randomly selected patients, 4 will
recover?

10. EXAMPLE 3:
In the old days, there was a probability of 0.8 of
success in any attempt to make a telephone call.
(This often depended on the importance of the person
making the call, or the operator's curiosity!)
Calculate the probability of having 7 successes in 10
attempts.

12. EXAMPLE 4:
The ratio of boys to girls at birth in Singapore is quite high at
1.09:1.
What proportion of Singapore families with exactly 6
children will have at least 3 boys?
(Ignore the probability of multiple births.)
[Interesting and disturbing trivia: In most countries the
ratio of boys to girls is about 1.04:1, but in China it is
1.15:1.]

14. EXAMPLE 5:
A manufacturer of metal pistons finds that on the average, 12% of his
pistons are rejected because they are either oversize or undersize.
What is the probability that a batch of 10 pistons will contain
(a) no more than 2 rejects?
(b) (b) at least 2 rejects?

17. BINOMIAL FREQUENCY DISTRIBUTION
 If the binomial probability distribution is multiplied by
the number of experiment N then the distribution is
called binomial frequency distribution.
f(X=x)= N P(X=x)=N
𝑛
𝑥
qn
_x px

18. MEAN & VARIANCE:
Mean, µ = n*p
Std. Dev. s =
Variance, s 2 =n*p*q
Where
n = number of fixed trials
p = probability of success in one of the n trials
q = probability of failure in one of the n trials

19.  Binomial distributions describe the possible number of times that
a particular event will occur in a sequence of observations.
 They are used when we want to know about the occurrence of
an event, not its magnitude.
Example:
 In a clinical trial, a patient’s condition may improve or not. We
study the number of patients who improved, not how much better
they feel.
 Is a person ambitious or not? The binomial distribution describes
the number of ambitious persons, not how ambitious they are.
 In quality control we assess the number of defective items in a lot
of goods, irrespective of the type of defect.