Question

4. Suppose P(A) = 0.48 and P(B) = 0.62. a. Can A and B be mutually exclusive? Why or why not. b. If A and B are independent, then what is the probability of both A and B occurring? c. Can A and B be mutually exclusive events and be independent events at the same time?

Answer 1

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

$P(A\cupB ) = P(A)+P(B)-P(A\cap B)$

Note that P(A) + P(B) = 1.1. Since the probability is always a number between 0 and 1, it must happen that $P(A\cap B) >0$, otherwise, $A\cup B$ 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

$P(A\cap B) = P(A)P(B)$

c) Recall that if we have mutuallly exclusive events A, B, then we have that  $P(A\cap B)=0$, if they were independent, we would have the following

$P(A) P(B) =0$

which necesarilly implies that at least one of the two events has 0 probability. (THis is not the case)