Can you help me solve this

Question

P(a)=.3 p(b)=.4 and p(a or b)=.7 are events a and b mutually exclusive? 

Answer 1

P(a ∩ b) = P(a) + P(b) - P(a ∪ B) = 0.3 + 0.4 - 0.7 = 0.

 

So if the sample space is discrete, these two events are mutually exclusive.

 

Here, the "discrete" sample space is important, since here is a counterexample with a continuous sample space:

 

Let X ∼ N(0,1) be a standard normal random variable and define the two events as follows:

 

a: X ≤ 0

 

b: X ≥ 0

 

Then a ∩ b is not empty, since it is the event that X = 0.

 

However, P(a ∩ b) = P(X = 0) = 0.