Question

A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events.

A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, ‘the number is even,’ and B be the event, ‘the number is red’. Are A and B independent?

Answer 1

S = {1, 2, 3, 4, 5, 6}

Let A : The number is even = {2, 4, 6}

`=> P(A) = 3/6  = 1/2`

B: The number in Red = {1, 2, 3}

`=> P(A) = 3/6 = 1/2`  and A ∩ B = {2}

`=> P(A ∩ B) = 1/6`

So `P(A).P(B) = = 1/2 xx 1/2 = 1/4`

then `P(A).P(B) != P(A nn B)` 

So A and B are not independent

Answer 2

The sample space for this experiment is S = {1, 2, 3, 4, 5, 6}

⇒ n(S) = 6

Event A = {2, 4, 6}

⇒ n(A) = 3

and event B = {1, 2, 3}

⇒ n(B) = 3

Then (A ∩ B) = {2}

⇒ n(A ∩ B) = 1

∴ P(A) = `(n(A))/(n(S))= 3/6 = 1/2`

P(B) = `(n(B))/(n(S))= 3/6 = 1/2`

⇒ P(A) . P(B) = `1/2 xx 1/2 = 1/4`

and P(A ∩ B) = `(n(A ∩ B))/(n(S)) = 1/6`

∵ P(A ∩ B) ≠ P(A) . P(B) 

∵ `(1/6 ne 1/2. 1/2)`

∴ Hence, events A and B are not independent.