# Two fair dice are thrown.Let ‘A’ denote the event that first die shows an odd number and ‘B’denote the event that the second die shows a prime numbe .Show that the events A & B are...

Two fair dice are thrown.

Let ‘A’ denote the event that first die shows an odd number and ‘B’denote the event that the second die shows a prime numbe .Show that the events A & B are independent.

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### 2 Answers

The dice role is random. So each die consists of a set {1 .. 6}. Knowing what was rolled on one die provides no information on what was rolled on the second die. Pretend the die A is labeled {1 ..6 } while die B is labeled { A .. B }. There is no union between these two sets, so the event of rolling the die remains independent. So the answer to your question is that, yes events A and B are independent.

You can see this by examining the conditional probability. Two events are independent if:

P(A|B) = P(A)

P(A|B) = 1/2 = P(A)

P(B|A) = 1/2 = P(B)

If A and B were on the same die, the answer would be different.

The standard definition of independence is:

Two events A and B are independent if and only if Pr(A ∩ B) = Pr(A)Pr(B).

A is the set: {1,3,5} --> P(A) = 1/2

B is the set: {1,2,3,5} --> P(B) = 2/3

A ∩ B is the set : {1,3,5} --> P(A ∩ B) = 1/2

Pr(A ∩ B) ≠ Pr(A)Pr(B)

Therefore these events are not independent.

Event A : showing up of 1,3 or 5 in the 1st die .The probability = 3/6 =1/2.

Event B: The showing up of prime number in the 2nd dice : Primes are 2,3,5 = 3way out of 6 = 1/2.

The joint way in which odd number in the first and prime number in the 2nd die showing up could happen is in the following ways:

Event AB

(1,2), (1,3), (1,5)

(3,2), (3,3), (3,5)

(3,2), (3,3), (3,5)

So it is in 9 ways out of 6*6 = 36 ways.

Therefore P(AB) = 9/36 = 1/4.

But P(A) = 1/2 and P(B) = ( 1/2).

Thus P(AB) = P(A)*P(B) estblishing that the events are independent.