Answer:
Step-by-step explanation:
a) Two events A,B are said to be mutually exlusive if any of the following occurs.
- A and B have an empty intersection
- A= B^c and their union is the whole sample space
- P(A intersection B) =0
REcall the following formula
[tex] P(A\cupB ) = P(A)+P(B)-P(A\cap B)[/tex]
Note that P(A) + P(B) = 1.1. Since the probability is always a number between 0 and 1, it must happen that [tex]P(A\cap B) >0[/tex], otherwise, [tex]A\cup B[/tex] would have a probability greater than 1. Hence, A, B are not mutually exclusive.
b) by definition, if two events are indepent, we have that the probability of the intersection is equal to products of their probabilites. Or
[tex]P(A\cap B) = P(A)P(B)[/tex]
c) Recall that if we have mutuallly exclusive events A, B, then we have that [tex]P(A\cap B)=0 [/tex], if they were independent, we would have the following
[tex] P(A) P(B) =0[/tex]
which necesarilly implies that at least one of the two events has 0 probability. (THis is not the case)
