SOLUTION: You are given the probabilities below: P(a) = 0.25 P(B) = 0.30 P(C) = 0.55 P(A and C) = 0.05 P(b and C) = 0 P(B|A) = 0.48 A.) Are A and B independent events? Why or Why no

Question 430376: You are given the probabilities below:
P(a) = 0.25
P(B) = 0.30
P(C) = 0.55
P(A and C) = 0.05
P(b and C) = 0
P(B|A) = 0.48
A.) Are A and B independent events? Why or Why not?
B.) Find P(A and B)
C.) Find P (A or B)
D.) Find P(A|B)
E.) Are B and C mutually exclusive events? Why or why not?
F.) Are B and C independent events? Provide mathematical proof.
Found 2 solutions by robertb, jim_thompson5910:
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
A) A and B are not independent: P(A)*P(B)= 0.25*0.30 = 0.075. This is not equal to P(A and B) = P(B|A)*P(A) = 0.48*0.25 = 0.12.
B) P(A and B) = P(B|A)*P(A) = 0.48*0.25 = 0.12.
C) P(A or B) = P(A) + P(B) - P(A and B) = 0.25 + 0.30 - 0.12 = 0.43
D) P(A|B) = P(A and B)/P(B) = 0.12/0.30 = 0.40
E) B and C are mutually exclusive, because it is given that P(B and C)=0 (Definition of mutual exclusivity.)
F) P(B)*P(C) = 0.30*0.55 = 0.165
P(B and C) = 0
==> The two quantities are not equal, hence B and C are not independent.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

The tutor @robertb has a great solution. I don't have much, if anything, to add really except for the fact that if A and B were independent, then,

P(A | B) = P(A)
P(B | A) = P(B)

Both must be true for A and B to be independent. We're told that P(B) = 0.30 and P(B | A) = 0.48 which contradicts the second equation shown above.

Put another way: If A and B were independent, then either event would not alter the probability of the other. The notation P(B | A) indicates that we are given event A has happened. Saying P(B|A) = 0.48 is the same as saying "the probability event B happens is 0.48 given A has happened". But this is altered from P(B) = 0.30; therefore event A has changed the probability of event B's occurrence. Ultimately the two events are not independent.

All of this explanation applies to part A only

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