A fair coin is tossed 4 times.
The following events are defined:
A = Exactly 2 tosses are heads
B = The second toss is a head
Are events A and B independent?
The Attempt at a Solution
My workbook says the events are independent, but it's just seems so counterintuitive.
Events for A
HHTT HTHT HTTH
TTHH THHT THTH
Pr(A) = 6/16 = 3/8
Pr(B) = 1/2
Events for A ∩ B
HHTT THHT THTH
Pr(A ∩ B) = 3/16
Pr(A) x Pr(b) = (3/8) x (1/2) = 3/16
Thus Events A and B are independent by my textbooks definition.
Mathematically these events appear to demonstrate that they are independent as Pr(A) x Pr(b) = Pr(A ∩ B), but I just can't see how the occurrence of event B does not affect event A. Knowing that B occurs does not change the probability of A occurring as there are half as many favourable outcomes and also half as many total outcomes. So does this mean that even though event B's occurence affects the favourabe outcomes of A, it is still technically independent of A because the numerical probability is unchanged?
Any comments would be appreciated.