# What is independence of events (in case of tossing an unbiased coin 3 times)?

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 $$C$$?

• Does it also bother you that $D=\{HHT\}$ and $B$ are not independent events? – bof Jun 1 '19 at 7:54

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.