A fair die is thrown. If E is the event that 'the number appearing is a multiple of 3' and F be the event that the number appearing is even'. Then choose the correct option given below:

E and F are independent events.

The correct answer is Option 4: E and F are independent events.

A fair die has sample space: S = {1, 2, 3, 4, 5, 6}

Step 1: Define events

E = multiples of 3
E = {3, 6}

F = even numbers
F = {2, 4, 6}

Step 2: Find probabilities

P(E) = 2/6 = 1/3

P(F) = 3/6 = 1/2

So the statement P(F) = 1/3 is false.

Step 3: Find 

E ∪ F = {2, 3, 4, 6}

P(E ∪ F) = 4/6 = 2/3

So P(E ∪ F) = 1/2 is false.

Step 4: Check if events are mutually disjoint

E ∩ F = {6}

Since the intersection is not empty, they are not mutually disjoint.

Step 5: Check independence

P(E ∩ F) = 1/6

P(E)P(F) = (1/3)(1/2) = 1/6

Since

P(E ∩ F) = P(E)P(F)

E and F are independent events.