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We flip a fair coin (independently) three times. De fine the following two events: A = "the number of tails is odd" B = "the number of heads is even" True or false: The events A and B are independent. (Recall that 0 is even.)

The answer is False, but I don't logically understand why, because if we know that if B occurs then we know that A also occurs. Does that means they are independent?

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If the number of tails is odd, then since we are flipping the coin 3 times, the number of heads must be even. Therefore, A is the same event as B and so they are very much not independent.

More generally, A and B are independent if and only if $P(A\cap B) = P(A)P(B)$.

In this scenario, since $A\cap B = A$, they are independent if and only if $P(A)= P(A)P(A)$ which is only the case when $P(A) = 0$ or $1$. However, with the fair coin neither of these is possible and so the two events are dependent.

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