Question

In: Statistics and Probability

Two fair dice are tossed. Let A be the maximum of the two numbers and let...

Two fair dice are tossed. Let A be the maximum of the two numbers and let B be the absolute difference between the two numbers. Find the joint probability of A and B. Are A and B independent? How do you know?

Expert Solution

Following is the sample space when we roll two fair dice:

In each outcome of the for (a,b) = c,d, first number a shows outcome of first die, second number b shows outcome of second die and c shows absolute difference and fourth number d shows the maximum of dice.

Each of the 36 outcomes are equally likely so the probability of each outcome will be 1/36.

Following table shows the joint pdf of A and B:

A	B	P(A=a, B=b)
1	0	1/36
2	1	1/36
3	2	1/36
4	3	1/36
5	4	1/36
6	5	1/36
2	1	1/36
2	0	1/36
3	1	1/36
4	2	1/36
5	3	1/36
6	4	1/36
3	2	1/36
3	1	1/36
3	0	1/36
4	1	1/36
5	2	1/36
6	3	1/36
4	3	1/36
4	2	1/36
4	1	1/36
4	0	1/36
5	1	1/36
6	2	1/36
5	4	1/36
5	3	1/36
5	2	1/36
5	1	1/36
5	0	1/36
6	1	1/36
6	5	1/36
6	4	1/36
6	3	1/36
6	2	1/36
6	1	1/36
6	0	1/36

Now we need to combine same values to get the joint probability distribution:

				A
		1	2	3	4	5	6	P(B=b)
	0	1/36	1/36	1/36	1/36	1/36	1/36	6/36
	1	0	2/36	2/36	2/36	2/36	2/36	10/36
B	2	0	0	2/36	2/36	2/36	2/36	8/36
	3	0	0	0	2/36	2/36	2/36	6/36
	4	0	0	0	0	2/36	2/36	4/36
	5	0	0	0	0	0	2/36	2/36
	P(A=a)	1/36	3/36	5/36	7/36	9/36	11/36	1

If A and B are independent then following must be true for each values of A and B:

P(A=a, B=b) = P(A=a)P(B=b)

From above we have

P(A=1, B=0) = 1/36

P(A=1) = 1/36

P(B=0) = 6/36

Since P(A=1, B=0) is not equal to P(A=1)P(B=0) so A and B are not independent.

orchestra answered 2 years ago

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