Are "first die even" and "sum is seven" independent?

Question

You roll two fair six-sided dice, a red one and a blue one. Consider two events:

$A$ : the red die shows an even number.
$B$ : the two dice sum to 7.

Are events $A$ and $B$ independent? Show your reasoning, not just a yes/no.

Accepted Answer

Independence means P(A \cap B) = P(A)\,P(B). Compute each piece over the 36 equally likely outcomes. - P(A) = 	frac{1}{2} (three of six red faces are even). - P(B) = 	frac{6}{36} = 	frac{1}{6}: the sum-7 pairs are (1,6),(2,5),(3,4),(4,3),(5,2),(6,1). - A \cap B: sum is 7 and the red die is even. From the list, the red die is even in (2,5),(4,3),(6,1), exactly 3 outcomes, so P(A\cap B) = 	frac{3}{36} = 	frac{1}{12}. Check the product: P(A)\,P(B) = 	frac{1}{2}\cdot	frac{1}{6} = 	frac{1}{12}. It matches, so the events are independent. The intuition: whatever the red die shows, there is a…

Are "first die even" and "sum is seven" independent?

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