Question

Three fair dice colored red, blue and green are rolled.

1. What will you choose the sample space to be in this case? How many atomic events are there in the sample space? What probability distribution will you use to model this problem?

2. What is the probability that exactly two of the dice roll the same number?

3. What is the probability that all three dice roll distinct numbers?

4. What is the probability that at least two of the dice roll the same number?

Give explanations for all your answers and show all the steps involved.

Answer 1

1)Three fair, six-sided dice are rolled. One green, one red, and one blue. Find the probability that precisely two of the dice show the same number. There are 66 outcomes for the red die, 66 outcomes for the green and 66 outcomes for the blue. Hence by the Multiplication Principle there are 6×6×6=2166×6×6=216 outcomes of the experiment.

2) What is the probability that exactly two of the dice roll the same number?

There are 33 possibilities for the die that is to show a different number and 66possible numbers for this die. The remaining two dice must have the same number, but different from the first die, whence there are 55 possibilities for this number. Hence by the Multiplication Principle there are 3×6×5=90 outcomes where exactly two dice show the same number, and thus

P(precisely 2 the same)=90/216=5/12P(precisely 2 the same)=90/216=5/12.

3. What is the probability that all three dice roll distinct numbers?

n(E)=number of ways have same faces =6

P(E)=n(E)/n(S)=6/216=1/36

4)What is the probability that at least two of the dice roll the same number?

Probability that at least two have the same number = 1 - probability that all three numbers are different = 1 - (5/6)*(4/6) = 1 - 20/36 = 1- 5/9 = 4/9 = 0.444