Find the number of elements in the sample space whose outcomes are all possible sequences of three rolls of the die. (a) Find the probability of the event that exactly one of the colors that appears face up is red. (b) Find the probability of the event that at least one of the colors that appears face up is red.

Sample space = 216 elements

a) P = 25/72

b) P = 91/216

Step-by-step explanation:

If each die has six possible values, the number of possible outcomes for three dice is 6^3 = 216 possible outcomes

a)

If we want exactly one die to be red, we have that one die is red (probability of 1/6) and the other two are not red (probability of 5/6), and the red die can be any of the three dice (combination of 3 choose 1), so the probability is:

P = C(3,1) * (1/6) * (5/6) * (5/6) = 3 * 25/216 = 75/216 = 25/72

b)

To find the probability of at least one die is red, we can find the probability of all dice not being red, and then find the complement of this probability (1 - P).

The probability of one die not being red is 5/6, so we have:

probability of all dice not being red = (5/6)^3 = 125/216

probability of at least one die being red = 1 - 125/216 = 91/216