#### Questions

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?

#### Solution 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

#### Solution 2

Total number of outcomes = 6

*A *: the event "number obtained is even"

The outcomes in favour of the event *A* are 2, 4, 6.

Number of outcomes in favour of event *A* = 3

`:. P(A) = 3/6 = 1/2`

B : the event "number obtained is red"

The outcomes in favour of the event B are 1, 2, 3.

Number of outcomes in favour of event B = 3

`:. P(B) = 3/6 = 1/2`

So,

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

Now

*A ∩ B* : the event "number obtained is even and red"

The outcome in favour of the event *A ∩ B* is 2.

Number of outcomes in favour of event *A ∩ B* = 1

∴ P(A *∩ B*) = `1/6`

Since P(A ∩ B) ≠ P(A)P(B), therefore, the events *A* and *B* are not independent events.