Page 52 - DMTH404_STATISTICS

Statistics

Notes
=   q
p
r i s r
r s
= P(A ) × P(A ).
1 2
Thus, the multiplication rule ‘is valid not only for individual sample points of S × S but also for
1 2
events in the component sample spaces S and S also. Here we have talked about events related
1 2
to two experiments. But we can easily extend this fact to events related to three or more
experiments.
The independent Bernoulli trials provide the simplest example of repeated independent trials.
Here each trial has only two possible outcomes, usually called success (S) and failure (F). We
further assume that the probability of success is the same in each trial, and therefore, the the
probability of failure is also the same for each trial. Usually we denote the probability of success
by p and that of failure by q = 1 – p.

Suppose, we consider three independent Bernoulli trials. The sample space is the Cartesian
product (S, F) × (S, F) × (S, F). It, therefore, consists of the eight points
SSS, SSF, SFS, FSS, FFS, FSF, SFF, FFF.

In view of independence, the corresponding probabilities are
p , p q, p q, p q, pq , pq , pq , q .
2
2
3
2
2
2
2
3
Do they add up to one? Yes.
In general, the sample space corresponding to n independent Bernoulli trials consists of 2 n
points. A generic point in this sample space consists of the sequence of n letters, j of which are S
and n – j are F, j = 0.1, . . ., n. Each such point carries the probability p q , probability of successes
n-j
i
 n 
in n independent Bernoulli trials. We first note that there are   points with j successes and
j
 
(n – j) failures (we ask you to prove this in E27). Since each such point carries the probability
n-j
i
p q , the probability of j successes, denoted by b(j, n, p) is
 n 
j

b(j, n, p) =   p q n j , 0, 1 , ....n.
j
 
These are called binomial probabilities and we shall return to a discussion of this topic when we
discuss the binomial distribution in Unit 8.
 n 
Task Prove that there are   pints with j successes and (n – j) failures in n Bernoulli
j
trials.  
Now we bring this unit to a closk. But before that let’s briefly recall the important concepts that
we studied in it.

3.5 Summary

 We have acquainted you with the concept of conditional probability P(A | B) of a given
the event B.
P(A  B)
P(A|B) = , P(B) 0

P(B)

44 LOVELY PROFESSIONAL UNIVERSITY