Suppose we toss two fair dice Let A denote the event that th

Suppose we toss two fair dice. Let A denote the event that the sum of the dice is six and B denote the event that the first die equals four. Show that the events A and B are not independent. Are they disjoint? Justify your answer.

Solution

In the above table the each block has 1/36 probability of happening. From this we can calculate the probability of sum of the outcomes being 6.

P(A) = 5/36

P(B) = 6/36 = 1/6

P(A and B) = 1/36

P(A or B) = 11/36

If events A and B were independent, then P(A and B) = P(A)*P(B)

Which you can calculate from above data to confirm that isnt true. Thus A and B must not be independent.

If A and B are disjoint then P(A and B ) = 0. As we can see that it is not the case here. Thus A and B are not disjoint.

	die 1
	outcome	1	2	3	4	5	6
die 2	1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12