# Dice

In probability and statistics, dice are commonly used to construct simple experiments. When a fair, six-sided die is rolled, the sample space can be denoted as:

{1, 2, 3, 4, 5, 6}

Each of the six possible outcomes has an equal likelihood of occurring. Given that A represents any one of the six outcomes, its probability can be denoted as:

P(A) = 1/6 = 0.167

There are many different ways that a die (or multiple dice) can be used to construct probability experiments, and it is common to define these experiments and events using set notation. For example, given the possible outcomes of rolling a fair, six-sided die,

- A = rolling an odd number
- B = rolling a number greater than 4
- C = rolling a prime number

determine the outcomes that satisfy the following conditions, then calculate their probabilities for a single roll of a die.

- The union of A and B (A ∪ B)
- The intersection of B and C (B ∩ C)
- The union of A, B, and C (A ∪ B ∪ C)

i. There are 3 values that satisfy event A: {1, 3, 5}; there are 2 values that satisfy event B: {5, 6}. Thus:

A ∪ B = {1, 3, 5, 6}

ii. Two values satisfy event B, {5, 6}, and three values satisfy event C, {2, 3, 5}, so:

B ∩ C = {5}

iii. The outcomes that satisfy events A, B, and C are {1, 3, 5}, {5, 6}, and {2, 3, 5} respectively, so:

A ∪ B ∪ C = {1, 2, 3, 5, 6}

The probability of an event (E) occurring can be calculated using the formula:

Thus, the probabilities of the events above occurring can be computed as follows.

i. P(A ∪ B):

ii. P(B ∩ C):

iii. P(A ∪ B ∪ C):

There are many other ways that dice can be used to demonstrate simple probability experiments. Refer to the roll a die page for more examples.