When rolling two dice, which of the following events are independent of the event that the first die is 4:

Question

When rolling two dice, which of the following events are independent of the event that the first die is 4:

(a) The second is 2

(b) The sum is 6

(c) The sum is 7

(d) The sum is even

I thought only option (a) is independent but that wasn't the case, so want to find out which ones are independent and which ones are dependent along with their reasoning.

There are multiple correct answers.

Answer 1

As an example, also (d) is independent of the event "first die is 4"

In fact

$$\mathbb{P}(\text{Sum is even})=\frac{1}{2}=\mathbb{P}(\text{Sum is even}|\text{First die is 4})$$

Check the rest in the same way finding a "tricky" result...

Answer 2

It would seem the events b and c are not independent. So let's check. For example P(sum is 7 given first roll is 4) is 1/6. If by chance it were independent P(sum is 7 given first roll is 4) would have been the same as P(sum is 7) which is 1/6 thus indeed independent