A coin is loaded so that the PrH 23 and the PrT 13 Todd t

A coin is loaded so that the Pr(H) = 2/3 and the Pr(T) = 1/3 . Todd tosses this coin twice.
Let A, B be the events
A: The first toss is a tail
B: Both tosses are the same
Are A, B independent?

Solution

There are 4 outcomes: HH, HT, TH, TT.

Events A and B are independent if P(A and B) = P(A) x P(B)

Here, A: first toss is tail and

B: both tosses are the same.

A happens if we get either TH or TT and B happens when we get either HH or TT.

Given P(H) = 2/3 and P(T) = 1/3.

Then, P(A) = P(TH) + P(TT) = 1/3 x 2/3 + 1/3 x 1/3 = 3/9 = 1/3

P(B) = P(HH) + P(TT) = 2/3 x 2/3 + 1/3 x 1/3 = 5/9

A and B happens when we get TT, this implies P(A and B) = P(TT) = 1/3 x 1/3 = 1/9

Now, P(A) x P(B) = 1/3 x 5/9 = 5/27 which is not equal to P(A and B) = 1/9.

Hence, events A and B are not independent.