The formula $P(E\cap F)=P (E)\times P (F)$ states that these events are independent. How is it that these events do not affect each others outcomes?

Question

A fair six sided dice is rolled twice.

$E=\{ \text{2 appears on the first roll}\}$

$F=\{\text{the sum on the two rolls is an even number}\}$

Using the formula $P(E\cap F)=P(E)\times P(F)$ the event are independent. I cannot understand this however as surely $E$ will affect the outcome of $F$. Could anyone explain how these are independent and how $E$ does not affect $F$?

$P(E)= \frac{1}{6}$ and $P(F)= \frac{18}{36}$. If $E$ occurred would it not affect $F$ outcome?

Answer 1

Since there is the same amount of even and odd numbers on a dice, it is independant. No mather what's rolled on the first dice, you have a 50% chance of having an even sum after you roll the second dice.

Answer 2

The answer is that there is a distinction between statistical independence and causal independence. Yes, whether the first roll is $2$, given what the second roll is, affects the value of $F$. But the question of statistical independence asks, does $P(F | E) = P(F|\neg E)$? The answer to that question is yes, and for the formula to hold true, all we need is statistical independence.