Question

let a, b, c be three events such that a, b are independent; b c are independent. does it imply that a, c are independent?

Answer 1

Answer:

No.

Step-by-step explanation:

To see that the answer is "no", we just need a counter-example.

Consider the case where the events a and c are actually the same event, while b is some event independent from this one.

Then:

• a and b are independent... check!
• b and c are independent... check!
• But a and c are certainly not independent as they are the same thing!

For a more concrete counter-example, take:

• a is the event "my coin toss results in a head"
• b is the event "your die roll results in a 3"
• c is the event "my coin toss results in a tail"

Then:

• P(a∩b) = 1 / 12 = P(a) × P(b)  =>  a and b are independent
• P(b∩c) = 1 / 12 = P(b) × P(c)  =>  b and c are independent
• P(a∩c) = 0 ≠ P(a) × P(c)  =>  a and c are dependent