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Exercise 1.44. Two fair dice are rolled. Let X be the maximum of the two numbers and Y the minimum of the two numbers on the dice. (a) Find the possible values of X and the possible values of Y . (b) Find the probabilities P(X ≤ k) for all integers k. Find the probability mass function of X. Hint. Noticing that P(X = k) = P(X ≤ k) − P(X ≤ k − 1) can save you some work. (c) Find the probability mass function of Y .

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Answer:

a. The possible values of X are 2,3,4,5,6

   The possible values of Y are 1,2,3,4,5

b.  X  1 2 3 4 5 6 Total

 n            0      2 4 6 8 10 30

  p(x) 0     1/15  2/15 3/15  4/15  1/3  1  

c.   X 1 2 3 4 5 6 Total

     n 10 8 6 4 2 0 30

  p(x)       1/3  4/15   3/15 2/15  1/15  0  1  

Step-by-step explanation:

Rolling two fair dice, we obtain the following tables :

 1 2 3 4 5 6

1 1,1 1,2 1,3 1,4 1,5 1,6

2 2,1 2,2 2,3 2,4 2,5 2,6

3 3,1 3,2 3,3 3,4 3,5 3,6

4 4,1 4,2 4,3 4,4 4,5 4,6

5 5,1 5,2 5,3 5,4 5,5 5,6

6 6,1 6,2 6,3 6,4 6,5 6,6

Let X be the maximum of the two numbers on the dice, then

X = {2,3,4,5,6,2,3,4,5,6,3,3,4,5,6,4,4,4,5,6,5,5,5,5,6,6,6,6,6,6}

Let Y be the minimum of the two numbers on the dice, then

Y = {1,1,1,1,1,1,2,2,2,2,1,2,3,3,3,1,2,3,4,4,1,2,3,4,5,1,2,3,4,5}

Finding the probability mass functions of X and Y which contains the possible values of X and Y and their associated probabilities, we have

X 1 2 3 4 5 6 Total

n 0 2 4 6 8 10 30

p(x) 0 1/15   2/15 3/15  4/15  1/3  1  

and

X 1 2 3 4 5 6 Total

n 10 8 6 4 2 0 30

p(x)  1/3 4/15  3/15 2/15  1/15  0  1 Â