probability

1. BINOMIAL
PROBABILITY
DISTRIBUTION

2. BASICS & TERMINOLOGY:
> Outcome:- The end result of an experiment.
> Random experiment
predictable.
> Random Event:- A
Experiments whose outcomes are not
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

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?

7. This is a binomial distribution because there are only 2 possible outcomes (we get a 5
or we don’t).
Now,;/ = 3 for each part. Let A" = number of fives appearing.
(a) Here, Y = 0.
P(X= 0) = cip'q-* = Co(i)°(f) = 2ll = ° 5787
(b) Here, x= 1.
^(Y- 1) = Clp'q”-* - C?(-J-)1(-|)2 ~ JY6 ~ 0.34722
(c) Here, .v = 3.
3 0
P(Y= 3) = C”xp*q"~* = C | ( i ) ( - J ) ' = 2 1 6 = 4 . 6296 x 10"3

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?

9. This is a binomial distribution because there are only 2
outcomes (the patient dies, or does not).
Let X = number who recover.
Here, // = 6 and ,v = 4. Letp = 0.25 (success - i.e. they live), <y
= 0.75 (failure, i.e. they die).
The probability that 4 will recover:
PLY) =
= C6
a(
0.25)4(0.75)2 =
15x 2.1973
xlO“3
0.0329595

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.

11. Probability of success p = 0.8,
so q- 0.2. X=success in
getting through.
Probability of 7 successes in
10 attempts:
Probability = P(X = 7)
= Cj°(0.8)7 (0.2)10-7 = 0.20133

13. 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. he probability of getting a boy is j QQ — 0. 521 5
et X = number of boys in the family
ere,
n = 6 ,
p = 0 5215,
<7 = 1 - 0 52153 = 0 4785
- 3 P(X) - Cn
xp*q- C|(0.5215)3(0.4785)3 - 0.31077
- 4 P(X) - C5(0.5215)4(0.4785)2 - 0.25402
- 5 P(X) - Cf(0.5215)5(0.4785)1 - 0. 1 1074
- 6 P(X) - C|(0.5215)6 (0.4785)° - 2. 0 1 1 5 x 10'2
o the probability of getting at least 3 boys is:
Probability = P(X >3)
- 0.31077+0.25402 + 0. 11074 + 2. Oil 5x 102
- 0.69565
NOTE: We could have calculated it like this:
P{X > 3) - 1 - (PC*0) + P(*,) + P(*2))

15. 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?

16. Let X = number of rejected pistons (In this case,
"success” means rejection!)
Here, n = 10, p = 0.12, q = 0.88.
(a)
No rejects
x - 0 P(X) - C”p*q*~x - cj°(0. 12)0(0.88)'°
One reject
AT - 1 P(X) - C|°(0. 12)'(0.88)® - 0.37977 Two rejects
x - 2 P(X) - Cj°(0. I2)2(0. 88)S - 0.23304 So the probability
of getting no more than 2 rejects is:
Probability = P(X < 2)
= 0.278 5+ 0.3 797 7 4-0.233 04

17. ptoceed 3$ Mows.
probabilityofatleast 2 reject*
.t-(PW+PW)
3 -(0W+0.3WT)
- 0.24H2

18. 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
v.
n
x
qn x px

19. MEAN & VARIANCE:
Mean, p = n*p
Std. Dev. s = V ” * P
Variance, s2 =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

20. Applications for binomial distributions
> 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
^^iaaQds,irrespective of the type of defect.