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An unbiased coin is tossed 3 times in a row. Define the events $A, B, C$ such that $A = \{HHH, TTT\}, B =\{TTT, TTH, THT, HTT\}$ and $C = \{HHH, TTT, HHT\}$

Now clearly $P(A\cap B) = P(A)P(B)$ and $P(C \cap B) \ne P(C)P(B)$ , so $A$ and $B$ are independent events while $C$ and $B$ are not.

I am really getting confused by the fact that, adding another elementary event$(\notin B)$ in A makes it dependent to B. Why? How does occurrence of the event $B$ does not depend on $A$ but depends on $C$?

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  • $\begingroup$ Does it also bother you that $D=\{HHT\}$ and $B$ are not independent events? $\endgroup$ – bof Jun 1 '19 at 7:54
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If you want a bit of intuition, think like this: if you know $A$ happened you have no idea if $B$ happened or not. With probability half you got $TTT$ and then $B$ happened and with the same probability you got $HHH$ and then $B$ didn't happen. So you have no idea, the probability of $B$ happening is exactly the same as it was if you didn't know that $A$ happened.

Now, with $C$ it is different. The probability of the event $B$ in general is $\frac{1}{2}$. However, if you know $C$ happened then with probability $\frac{2}{3}$ you know that you got either $HHT$ or $HHH$. So the probability that $B$ did happen is not more than $\frac{1}{3}$ in that case. There is dependence indeed.

Anyway, this is just intuition. There are formal definitions for independence.

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