A question states that a fair coin is tossed thrice, and these events are defined; $A$: the first toss is a head; $B$: the second toss is a head; $C$: exactly two consecutive heads or exactly two consecutive tails occur.
That all three are pairwise independent is obvious. However, apparently they're also mutually independent.
How does that work? As I understand it, if the knowledge of two events here affects the chances of another, the events wouldn't be mutually independent. Here, if you know the first and second tosses to both yield heads, the probability of C would be 1.
So how are they mutually independent?
Best Answer
If you know $A$ and $B$ occur, then you still don't know whether $C$ occurs, because the third toss could be heads or tails. The probability isn't $1$ because it says exactly two consecutive. No matter how $A$ and $B$ turn up, there will always be one way to achieve exactly two consecutive.
It appears $P(A \cap B \cap C) = 1/8 = P(A)P(B)P(C) = 1/2\cdot 1/2 \cdot 1/2.$. Together with the pairwise independence, the three events are mutually independent.