Three different dice are thrown three times. What is the probability that they show different numbers only two times?
Three different dice are thrown three times. What is the probability that they show different numbers only two times?
Let's assume that these are three unbiased, six-sided dice. Also, we'll assume that the OP means to find the probability that they show three different numbers on exactly two out of three rolls.
First, we need the probability that the three dice are all different numbers on a single roll of the three dice:
Regardless of the number rolled on the first die, the probability that the number rolled on the second die is different from the first die is 5/6. The probability that the number rolled on the third die is different from the first two dice is 4/6. Therefore, the probability that the three dice are all different on a single roll is 5/6 * 4/6 = 20/36 = 5/9.
Now, we need the probability that all three dice will be different on exactly two out of the three rolls:
If the probability of all three dice being different on any one roll is 5/9, then the probability of one or more dice matching on that one roll is 1 - 5/9 = 4/9.
The probability that we get a match on the first roll but all different numbers on the second and third rolls is (4/9)*(5/9)*(5/9) = 100/729.
The probability of getting a match on just the second roll, or on just the third roll, is the same: 100/729.
Therefore, the probability of a match on just one roll (and therefore, all different numbers of two of the three rolls as the question asked), is:
3 * (100/729) = 100/243 = 0.41152263.
Let's assume that these are three unbiased, six-sided dice. Also, we'll assume that the OP means to find the probability that they show three different numbers on exactly two out of three rolls.
First, we need the probability that the three dice are all different numbers on a single roll of the three dice:
Regardless of the number rolled on the first die, the probability that the number rolled on the second die is different from the first die is 5/6. The probability that the number rolled on the third die is different from the first two dice is 4/6. Therefore, the probability that the three dice are all different on a single roll is 5/6 * 4/6 = 20/36 = 5/9.
Now, we need the probability that all three dice will be different on exactly two out of the three rolls:
If the probability of all three dice being different on any one roll is 5/9, then the probability of one or more dice matching on that one roll is 1 - 5/9 = 4/9.
The probability that we get a match on the first roll but all different numbers on the second and third rolls is (4/9)*(5/9)*(5/9) = 100/729.
The probability of getting a match on just the second roll, or on just the third roll, is the same: 100/729.
Therefore, the probability of a match on just one roll (and therefore, all different numbers of two of the three rolls as the question asked), is:
3 * (100/729) = 100/243 = 0.41152263.
