A 6-sided number cube, with faces numbered 1 through 6, is to be rolled twice. What is the probability that the number that comes up on the first roll will be less than the number that comes up on the second roll?
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Note that since there are 6 possible outcomes to a roll of a six-sided number cube, there are 6^2 = 36 ways to roll a six-sided number cube twice.
Now, consider the cases where the second roll was a 1, 2, 3, 4, 5, or a 6 respectively.
Suppose that the second roll was a 1. There is no way to roll less than this. Next, suppose that you roll a 2 on the second time. In order to roll less than 2 on the first roll, you need to roll a 1; this gives 1 way to roll less than a 2. In a similar fashion, you need to a roll a 1 or a 2 for the first roll to roll less than a 3 (2 ways); a 1, 2, or a 3 to roll less than a 4 (3 ways); a 1, 2, 3, or a 4 to roll less than a 5 (4 ways); or a 1, 2, 3, 4, or a 5 to roll less than a 6 (5 ways). Thus, there are 1 + 2 + 3 + 4 + 5 = 15 ways to roll a six-sided number cube twice such that the first roll's number was less than the second.
Therefore, the required probability is 15/36 = 5/12.
I hope this helps!
Now, consider the cases where the second roll was a 1, 2, 3, 4, 5, or a 6 respectively.
Suppose that the second roll was a 1. There is no way to roll less than this. Next, suppose that you roll a 2 on the second time. In order to roll less than 2 on the first roll, you need to roll a 1; this gives 1 way to roll less than a 2. In a similar fashion, you need to a roll a 1 or a 2 for the first roll to roll less than a 3 (2 ways); a 1, 2, or a 3 to roll less than a 4 (3 ways); a 1, 2, 3, or a 4 to roll less than a 5 (4 ways); or a 1, 2, 3, 4, or a 5 to roll less than a 6 (5 ways). Thus, there are 1 + 2 + 3 + 4 + 5 = 15 ways to roll a six-sided number cube twice such that the first roll's number was less than the second.
Therefore, the required probability is 15/36 = 5/12.
I hope this helps!
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Clearly the probability of doubles (both rolls matching) is 1/6, since the first roll can be anything and then the probability that the second roll matches it is 1/6.
Thus the probability of not getting doubles is 5/6. By symmetry, the probability that the second roll is less than the first roll equals the probability that the first roll is less than the second roll; the sum of these two equal probabilities is the probability of not getting doubles (which is 5/6).
Thus, the probability that the first roll is less than the second roll is (1/2)(5/6) = 5/12.
Lord bless you today!
Thus the probability of not getting doubles is 5/6. By symmetry, the probability that the second roll is less than the first roll equals the probability that the first roll is less than the second roll; the sum of these two equal probabilities is the probability of not getting doubles (which is 5/6).
Thus, the probability that the first roll is less than the second roll is (1/2)(5/6) = 5/12.
Lord bless you today!
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This might be easier to see by listing.
There are 6*6= 36 possible combinations
11, 12, 13, 14, 15, 16
There are 6*6= 36 possible combinations
11, 12, 13, 14, 15, 16
12
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