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Two ordinary fair dice (one red and one blue) are thrown.

Event A: The red die will show a 5 or a 6.

Event B: The sum of the two dice will be 7.

Event C: The sum of the two dice will be 8.

Using the test for independence: $$P(A\cap B) = P(A).P(B) $$ It can be seen that $A$ and $B$ are independent events whilst events $A$ and $C$ are not independent.

I can't understand why this is the case. I understand the mathematics but I can't understand the logic behind it. I drew out the sample space but I am none the wiser. Is there some intrinsic difference between events $B$ and $C$ that results in one being independent and the other not?

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If the red die is 5 or 6 it doesn't reduce or increase the chance of the sum being 7 because every value the red die could be could lead to a sum of 7 with equal chance.

This can't be said of a sum of 8 though. Because if the red die would have been a 1, a sum of 8 is now impossible. By saying that the red die is a 5 or 6, we are increasing the chance of rolling an 8.

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    $\begingroup$ This highlights that it is helpful to think of independence $P(A \cap B) = P(A) P(B)$ as $P(A \cap B) / P(B) = P(A)$, i.e. that the conditional probability is the same as the original probability. (When $P(B) \not = 0$ clearly.) $\endgroup$ – Lorenzo Najt Sep 16 '16 at 13:13
  • $\begingroup$ Thank you turkeyhundt and AreaMan. I now understand. $\endgroup$ – Kantura Sep 16 '16 at 13:22

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