An experiment has 10 equally likely outcomes. Let and be two non-empty events of the experiment. If consists of 4 outcomes, then the number of outcomes that must have, so that and are independent, is
- A.2, 4 or 8
- B.3, 6 or 9
- C.4 or 8
- D.5 or 10
Show answer and solution
Correct option: D
Solution
P(A) = 4/10 = 2/5. For independence, P(A∩B) = P(A)·P(B) = (2/5)·(|B|/10) = |B|/25. Since P(A∩B) must equal |A∩B|/10 for some non-negative integer |A∩B|, we need |B|/25 = |A∩B|/10, i.e., |A∩B| = 2|B|/5. For this to be a positive integer, |B| must be a multiple of 5. Since B is non-empty and has at most 10 outcomes, |B| = 5 or 10. Both are achievable: if |B|=5, then |A∩B|=2; if |B|=10, then |A∩B|=4.
[AIEEE 2008]