Question

1 . An experiment has four equiprobable outcomes. S { 2.3) and C { 1 , 2.3.4). We define three events A { 1 . 2 }. В { 1.3). Are A. B and C independent? Justify your answer.

Answer 1

Two events are said to be independent if the occurrence of one event does not affect the occurrence of the other. The probability of occurring of the two events are independent of each other.

Statistically, an event A is said to be independent of another event B, if the conditional probability of A given B is equal to the unconditional probability of A i.e. P(A | B) = P(A)

Also, the events A and B are independent if P(A ∩ B) = P(A) P(B)

In this context

S= {1,2,3,4}

Event A = {1,2}

Event B= {2,3}

Event C= {1,3}

P(A) = 2/4 = 1/2

P(B)= 2/4 = 1/2

P(C)= 2/4 = 1/2

And

A ∩ B = {2}

$\fn_cm \Rightarrow$P(A ∩ B)= 1/4

A ∩ C = {1}

$\fn_cm \Rightarrow$P(A ∩ C)= 1/4

B ∩ C = {3}

$\fn_cm \Rightarrow$P(B ∩ C)= 1/4

To prove A independent B ,consider

P(A ∩ B) = P(A). P(B)

L.H.S = P(A ∩ B) = 1/4

R.H.S = P(A) .P(B) = 1/2 *1/2 = 1/4

$\fn_cm \Rightarrow$ L.H.S = R.H.S

Hence event A and B are independent

Similarly, consider

P(A ∩ C) = P(A). P(C)

L.H.S = P(A ∩ C) = 1/4

R.H.S = P(A) .P(C) = 1/2 *1/2 = 1/4

$\fn_cm \Rightarrow$ L.H.S = R.H.S

Hence event A and B are also independent.

So, event A ,B and C are independent