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?

Solutions

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 1 year ago

Next > < Previous

Related Solutions

Two fair six-sided dice are tossed independently. Let M = the maximum of the two tosses...

Two fair six-sided dice are tossed independently. Let M = the maximum of the two tosses (so M(1,5) = 5, M(3,3) = 3, etc.). (a) What is the pmf of M? [Hint: First determine p(1), then p(2), and so on.] (Enter your answers as fractions.) m 1 2 3 4 5 6 p(m) (b) Determine the cdf of M. (Enter your answers as fractions.)F(m) = m < 1 1 ≤ m <...

Two fair dice are tossed together. Let X be the sum and Y the product of...

Two fair dice are tossed together. Let X be the sum and Y the product of the two numbers on the top of the dice. Calculate E(X+ 3Y).

7. (Sec. 3.2) Two fair six-sided dice are tossed independently. Let M = the minimum of...

7. (Sec. 3.2) Two fair six-sided dice are tossed independently. Let M = the minimum of the two tosses. For example, M(2, 5) = 2, M(4, 4) = 4, etc. (a) What is the PMF of M? [Hint: just work out each probability individually by counting the number of outcomes which result in a specific value for M, i.e. find p(1), then p(2), and so on up to p(6)]. (b) Determine the CDF of M. ( c) Graph the CDF...

Q2. Two fair dice are tossed and recorded (a) What is the probability that the sum...

Q2. Two fair dice are tossed and recorded (a) What is the probability that the sum of the two dice is at most 10? (b) Given that the sum is an even number, what is the probability that the sum of two dice is 6 or 10?

You roll two fair dice. Let A be the event that the sum of the dice...

You roll two fair dice. Let A be the event that the sum of the dice is an even number. Let B be the event that the two results are different. (a) Given B has occurred, what is the probability A has also occurred? (b) Given A has occurred, what is the probability B has also occurred? (c) What is the probability of getting a sum of 9? (d) Given that the sum of the pair of dice is 9...

Two fair dice are tossed, and the up face on each die is recorded. Find the...

Two fair dice are tossed, and the up face on each die is recorded. Find the probability of observing each of the following events: A: { The sum of the numbers is even } B: { A 4 appears on at least one of the dice } C: { The difference of the numbers is 2 or less }

Two fair dice are tossed, and (X,Y) denote the number of spots on the first and...

Two fair dice are tossed, and (X,Y) denote the number of spots on the first and on the second dice. Consider two random variables: U = X + Y and W = | X - Y |. A). Derive the distribution of U. List all possible values and evaluate their probabilities. B). Derive the distribution of W. List all possible values and evaluate their probabilities. C). Determine the conditional probability P[6 <= U <= 7 | W <= 1]

Two fair dice are tossed, and the face on each die is observed. Use a tree...

Two fair dice are tossed, and the face on each die is observed. Use a tree diagram to find the 36 sample points contained in the sample space. Assign probabilities to the sample points in part a. Find the probability of each of the following events: A = {3 showing on each die} B = {sum of two numbers showing is 7} C = {sum of two numbers showing is even} Also, for events A, B, and C, which one...

suppose two dice are tossed. let x= the sum of the on the top faces of...

suppose two dice are tossed. let x= the sum of the on the top faces of the two dice. find the probability distribution of X

Two fair dice, one blue and one red, are tossed, and the up face on each...

Two fair dice, one blue and one red, are tossed, and the up face on each die is recorded. Define the following events: E:E: {{ The sum of the numbers is even }} F:F: {{ The difference of the numbers is 3 or more }} Find the following probabilities: P(E)= P(F)= P(EandF)= P(E|F)= P(F|E)=

Subjects