Discrete Probability Distributions

Question

Create the probability distribution for each random variable described.

The number of even numbers showing when a pair of standard six-sided dice are rolled.

Answer 1

There are 36 possible combinations of rolls, each equally likely:

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65 66

Among these, we can either have no even numbers (i.e., both numbers are odd), one even number (and one odd) or two even numbers.

Let's let X be the number of even numbers. Then:

P(X=0) = number of rolls with no even numbers / 36 (total number of possible rolls)

There are nine combinations with no even numbers (11, 13, 15, 31, 33, 35, 51, 53, 55), so P(X=0) = 9 / 36 = 1/4.

P(X=1) = number of rolls with exactly one even number / 36

There are 18 combinations with exactly one even number (12, 14, 16, 21, 23, 25, 32, 34, 36, 41, 43, 45, 52, 54, 56, 61, 63, 65), so P(X=1) = 18 / 36 = 1/2.

And finally, P(X=2) = number of rolls with two even numbers / 36 and there are 9 rolls with two even numbers (22, 24, 26, 42, 44, 46, 62, 64, 66), so P(X=2) = 9 / 36 = 1/4.

So we have:

X P(X=x)

0 1/4

1 1/2

2 1/4