Event A has probability 0.4 to occur and Event B has probability 0.7 to occur.

Are a and B mutually exclusive ( disjoint)? Explain.

If A and B are independent, what is the probability P(A or B^c) ?

No. If they were disjoint, they would have to sum to less than 1. Think about a spinner from a board game, where the possible results are from 1 to 10:


Result A =you get 1, 2, or 3; which has a 30% chance
Result B = you get 4,5,6,7,8 which has a 50% chance
A and B are disjoint because any result you get can AT MOST only fall into one of those two categories.
Also we need to be careful here. Since I gave you the results we can say for sure they are disjoint. However if I had just said A=30% B= 50% we can't say for sure if they are disjoint or not.

Now lets try to set it up with C = 40% and D = 70%
Result C =you get 1, 2, 3, or 4; which has a 40% chance
Result D = you get 4,5,6,7,8,9,or 10 which has a 70% chance
A and B are not disjoint, since there is a result (4) which falls into both.
Here, even if we were not given the results, we could still say they are not disjoint, since there is not way to divide it up without some overlap.

So if the two categories sum to more than 100%, there will be some overlap.

This post was edited by Azrad on Apr 12 2013 12:09pm

A and B independant : p(A and B ) = p(A) x p(B), and p(A and B^c) = p(A) x p(B^c)

p(A or B^c) = p(A) + p(B^c) - p(A and B^c)
p(A or B^c) = p(A) + 1 - p(B) - p(A) x p(B^c)
p(A or B^c) = p(A) + 1 - p(B) - p(A) x (1 - p(B))
p(A or B^c) = 1 - p(B) + p(A) x p(B)

which gives 0.58 in your example.

This post was edited by feanur on Apr 12 2013 01:06pm