Probability Question

Question

A fair coin is tossed 3 times. Events A and B are defined by A: The 2nd toss is a tail; B: The 1st and last tosses are the same. Are A and B independent? Why and how?

Answer 1

They are independent if p(A and B) = p(A).p(B)

p(A) = 1/2

p(B) : There are 2 x 2 x 2 = 8 possible outcomes. If first and last are the same we can have

H H H

H T H

T H T

T T T

so 4 possible outcomes out of 8, so p(B) = 4/8 = 1/2

p (A and B) . Possible outcomes. HTH TTT, so p(A and B) = 2/8 = 1/4

p(A).p(B) = p(A and B) so A, B are independent.

Because A is about the 2nd toss , and B is about the other tosses (1st and 3rd) so knowing A has happened or B has happened doesn;t change the probability of the other happening