Probability Proof

**weakmath** · September 15th 2008, 03:03 PM

Suppose that A and B are not trivial events, that is 0< P(A) < 1 and 0 , P(B) < 1. show that

a. if A subset B, then A and B are not independent;

b. if B subset A, then A and B are not independent;

c. if A and B are disjoint then A and B are not independent

**Plato** · September 15th 2008, 03:27 PM

Have you noticed that parts a & b are identical?
a) $P(B|A) = \frac{{P(B \cap A)}}{{P(A)}} = \frac{{P(A)}}{{P(A)}} = 1 \ne P(B)$ .
Do you see a contradiction therein if they were independent?

$A \cap B = \emptyset \Rightarrow \quad P(A \cup B) = P(A) + P(B) - P(A \cap B) \Rightarrow \quad P(A \cap B) = 0$
Another contradiction???

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