Two events A and B are said to be independent of one another, if
P(Aâˆ©B) = P(A) Ã— P(B).

**Independent
Events**** **

For
any two events *A* and *B* of a random experiment, if *P*(*A*/*B*)
= *P*(*A*), then knowledge of the event *B*
does not change the probability for the occurrence of the event *A*.
Such events are called independent events.

If
P(A/B) = P(A), then

=> P(Aâˆ©B) = P(A) Ã— P(B).

Similarly,
the relation *P*(*B*/*A*) = *P*(*B*) also indicates the independence
of the events *A *and* B.*

* Two events A and B are said to be independent
of one another, if*

* P(Aâˆ©B) = P(A) Ã— P(B).*

In
tossing a fair coin twice, let the events *A* and *B* be defined as *A*:
getting head on the first toss, *B*: getting head on the second toss.
Prove that *A* and *B* are independent events.

The
sample space of this experiment is** **

*S = {HH, HT, TH, TT}.*

The
unconditional probabilities of *A* and *B* are P(*A*) = 1/2= P(*B*).

The event of getting heads in both the tosses is represented by
A** âˆ©**B.
The outcome of the experiment in favour of the occurrence of this event is HH.
Hence, P(A

P(A** âˆ©**B) = P(A) Ã— P(B) holds.
Thus, the events A and B are independent events.

In the experiment of rolling a pair of dice, the events A, B and
C are defined as *A *: getting 2 on the
first die,* B *: getting 2 on the
second die, and* C *: sum of the faces
of* *dice is an even number. Prove that
the events are pair wise independent but not mutually independent?

S= { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6),

(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),

(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),

(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),

(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),

(6,1), (6,2), (6,3), (6,4), (6,5), (6,6) }

n(S) = 36

The outcomes which are favourable to the occurrence of these
events can be listed below:

A = { (2,1), (2,2), (2,3), (2,4), (2,5), (2,6) }

n(A) = 6

P(A) = 6/36=1/6

B = { (1,2), (2,2), (3,2), (4,2), (5,2), (6,2) }

n(B) = 6

P(B) = 6/36=1/6

C = { (1,1), (1,3), (1,5), (2,2), (2,4), (2,6), (3,1), (3,3),
(3,5),

(4,2), (4,4), (4,6), (5,1), (5,3), (5,5), (6,2), (6,4), (6,6) }

n (C) = 18

P(C) = 18/36=1/6

Aâˆ©B = { (2,2) }

n(Aâˆ©B ) = 1

P(Aâˆ©B) = 1/36

The following relations may be obtained from these probabilities** **

The above relations show that when the events *A*, *B*
and *C* are considered in pairs, they
are independent. But, when all the three events are considered together, they
are not independent.

Tags : Probability Theory , 11th Statistics : Chapter 8 : Elementary Probability Theory

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11th Statistics : Chapter 8 : Elementary Probability Theory : Independent Events | Probability Theory

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