Suppose you have two six-sided dice where each side is equally likely to land face up when rolled. What is the probability that you will roll doubles? What is the probability that you will roll a sum of four? Are these empirical or a theoretical probabilities?

16.7% probability that you will roll doubles.

8.3% probability that you will roll a sum of four.

Theoretical probabilities;

Step-by-step explanation:

These are all the possible outcomes:

(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)

There are 36 possible outcomes

What is the probability that you will roll doubles?

There are 6 outcomes that are doubles. (1,1), (2,2), (3,3), (4,4), (5,5), (6,6)

So there is a 6/36 = 1/6 = 0.166 = 16.7% probability that you will roll doubles.

What is the probability that you will roll a sum of four?

There are 3 outcomes with a sum of four: (1,3), (2,2), (3,1)

So there is a 3/36 = 0.083 = 8.3% probability that you will roll a sum of four.

Are these empirical or a theoretical probabilities?

Empirical probability is about the observed counts, it is calculated after the dice has been thrown.

Theoretical are the expected probabilities, calculated before the event happens. In this case, these are theoretical probabilities.