Probability Distribution question for a die

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Aurelius032

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We have a die that is numbered 1 – 3 with two 1’s, two 2’s, and two 3’s. Let’s assume it is fair (each side is equally probable independent.) and that the die is rolled 3 times. And, let’s assume that the 3 rolls are Let X, Y, and Z be the outcomes of the first, second and third rolls, respectively.

a. What is the probability distribution of X+Y+Z? That is, create a table that contains each unique possible value of X+Y (each value only listed once) and each possibility’s corresponding probability.

What does this mean exactly? Am I suppose to list each unique combination out?

I.e. First Roll: X = Rolled, Y = Not Rolled, Z = Not Rolled
Second Roll: X = Rolled, Y = Not Rolled, Z = Not Rolled
Third Roll: X = Not Rolled, Y = Rolled, Z = Not Rolled

b. What is the probability that X+Y+Z is greater than or equal to 7?

How can the probability of this possibly be 1? I guess if we take the average product of outcomes it can be more than that.. but I just don't see how this is even possible.

I'm just looking for direction. Thank you!
 
Aurelius032 said:
We have a die that is numbered 1 – 3 with two 1’s, two 2’s, and two 3’s. Let’s assume it is fair (each side is equally probable independent.) and that the die is rolled 3 times. And, let’s assume that the 3 rolls are Let X, Y, and Z be the outcomes of the first, second and third rolls, respectively.

a. What is the probability distribution of X+Y+Z? That is, create a table that contains each unique possible value of X+Y (each value only listed once) and each possibility’s corresponding probability.

What does this mean exactly? Am I suppose to list each unique combination out?

I.e. First Roll: X = Rolled, Y = Not Rolled, Z = Not Rolled
Second Roll: X = Rolled, Y = Not Rolled, Z = Not Rolled
Third Roll: X = Not Rolled, Y = Rolled, Z = Not Rolled

b. What is the probability that X+Y+Z is greater than or equal to 7?

How can the probability of this possibly be 1? I guess if we take the average product of outcomes it can be more than that.. but I just don't see how this is even possible.

I'm just looking for direction. Thank you!
Click to expand...
This is my take on the question: X, Y, and Z can have values of 1, 2, or 3. Since repeats are allowed we have
63 possibilities if we color the numbers, for example red and yellow 1, etc., or 33 possibilities if we don't color the numbers and treat both 1's, both 2's, and both 3's the same. The out comes are
(x1,y1,z1) = (1, 1, 1); S1 = 1 + 1 + 1 = 3
(x2,y2,z2) = (1, 1, 2); S2 = 4
(x3,y3,z3) = (1, 1, 3); S3 = 5
...
(x27,y27,z27) = (3, 3, 3); S27 = 9


a. What is the distribution of S. That is what is the probability of some S=3, of S=4, of S=5, ..., of S=9.
b. What is the probability that S is greater than 7.
 
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