Question

You perform a random experiment by rolling one time two fair six-sided dice and record the number of each die. Define and/or give examples of the following:

a. Give an example for a single outcome of your experiment?
b. What is the sample space of your experiment?
c. What is the probability for the first die to be two?
d. What is the probability that the value rolled on die 1 minus the value rolled on die 2 is positive?

Answer 1

(c)1/6

(d)5/12

Explanation:

When two fair six-sided dice are rolled once

(a)The pair (x,y) denotes a single outcome of the experiment where x is the outcome of the first die and y is the outcome of the second die.

For example, (2,1) means the first die shows a 2 while the second die shows an outcome of 1.

(b)Sample Space

The sample space of all possible outcome is:

`(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)\\(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)\\(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)\\(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)\\(5, 1), (5, 2), (5, 3), (5, 4), (5, 5),(5,6)\\(6, 1), (6, 2), (6, 3), (6, 4), (6,5), (6,6)`

Total Number of Outcomes =36

(c)The probability for the first die to be two

The outcomes where the fist die is 2 are:

`(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)`

Therefore:

The probability for the first die is two

`=(6)/(36)\\\\=(1)/(6)`

(d)The probability that the value rolled on die 1 minus the value rolled on die 2 is positive

These are the outcomes of the pair (x,y) where x>y.

They are:

`(2, 1)\\(3, 1), (3, 2)\\(4, 1), (4, 2), (4, 3)\\(5, 1), (5, 2), (5, 3), (5, 4)\\(6, 1), (6, 2), (6, 3), (6, 4), (6,5)`

`P(x-y>0)=(15)/(36)\\\\=(5)/(12)`