probabilities of compound events

Questions:

Consider a single roll of a die. Define event A as getting a number greater than 3. Define event B as getting an even number. Are A and B mutually exclusive? Are A and B independent?
S = {1,2,3,4,5,6}
P(A) = 3/6 = 1/2
SA = {4,5,6}
P(B) = 3/6 = 1/2
SB = {2,4,6}

P(A and B) = 2/6 = 1/3

Two events are said to be disjoint, or mutually exclusive, if and only if P(A and B) = 0.

A and B are not mutually exclusive.

Two events are said to be independent if P(A and B) = P(A)P(B), provided that P(A) and P(B) are both nonzero.
P(A) = 1/2
P(B) = 1/2
P(A) * P(B) = 1/4
Since P(A and B) = 1/3
P(A) is not equal to P(A)P(B)


A and B are not independent.

Give an example of a pair of events A and B that are disjoint but not independent.

P(A) = rolling a dice and getting an odd number. = 3/6 = 1/2
P(B) = rolling a dice and getting an even number. = 3/6 = 1/2

SA = {1,3,5}
SB = {2,4,6}
P(A and B) = 0

P(A and B) is not equal to P(A)P(B) which is 1/4, so P(A) and P(B) are not independent.




* Give an example of a pair of events A and B that are independent but not disjoint.

Event A - Getting an odd number when a dice is rolled. Sample = {1,3,5}
Event B - Getting a 5 . Sample = {5}
P(A and B) is not zero so they are not mutually exclusive (disjoint)
Are these independent ? No. As P(A and B) = 1/6
P(A)*P(B) = 3/6 * 1/6 = 3/36 = 1/12
* Is it possible for events to be both independent and disjoint? Why or why not?

Events can not be both independent and mutually exclusive.
To be independent P(A and B) = 0
To be disjoint P(A and B) must be equal to P(A) P(B), where P(A) and P(B) are not equal to zero. which is not possible.

* If A and B are independent, do you think that Ac and B are necessarily independent?

Ac (Complement of event A) = 1 - P(A)

A and B are independent which means that P(A and B) = P(A) P(B)
Assuming the following is true:
P(Ac and B) = P(Ac) P(B) = (1 - P(A) )P(B)
P(B) - P(A) P(B) = P( A and B)/P(A)) - P(A)