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Ok, so here’s the answer. The key here is that order matters in the way you roll each triplet. Take for example the triplet 1,2,6. You can roll a 1 on the first roll, a 2 on the second roll, and a 6 on the third roll. Or, you can roll a number of different orders of the same three numbers. There are 3! (read “three factorial”) different orders for this triplet. You have three possibilities for the first roll, two for the second, and one for the last roll. So, you have 3!=6 possible ways to roll a sum of 9 that include a 1, 2, and 6.
Now, note that all the triplets do not havc unique rolls. Take for example the triplet 3,3,3. There is only one possible way to roll a 3, then a 3, and then a 3. What about 1,4,4? This is a permutation. You can have 1,4,4 or 4,1,4 or 4,4,1. There are three possibilities.
Now, what we do is add up the number of ways of rolling a 9 or a 10. There are 25 possible ways of rolling a 9 and 27 possible ways of rolling a 10. We could compare the number of possibilities and say that 10 is more likely, but that isn’t good form. We should compare the exact probabilities of rolling a sum of 9 or ten on exactly 3 rolls.
Now, how many outcomes for rolls are there when you toss the die three times? For the first time, it is 6. For the second time, you have 6 more possibilities, and for the final roll you have again, 6 possible outcomes. 6x6x6=216. There are 216 possible ways to roll in this demonstration. So, the probability of rolling a sum of 9 with exactly three rolls of the die would be 25/216 and the sum of 10 with three rolls would be 27/216. Thus, the probability of obtaining a sum of 10 in exactly 3 rolls is higher.
Any comments?
Now, note that all the triplets do not havc unique rolls. Take for example the triplet 3,3,3. There is only one possible way to roll a 3, then a 3, and then a 3. What about 1,4,4? This is a permutation. You can have 1,4,4 or 4,1,4 or 4,4,1. There are three possibilities.
Now, what we do is add up the number of ways of rolling a 9 or a 10. There are 25 possible ways of rolling a 9 and 27 possible ways of rolling a 10. We could compare the number of possibilities and say that 10 is more likely, but that isn’t good form. We should compare the exact probabilities of rolling a sum of 9 or ten on exactly 3 rolls.
Now, how many outcomes for rolls are there when you toss the die three times? For the first time, it is 6. For the second time, you have 6 more possibilities, and for the final roll you have again, 6 possible outcomes. 6x6x6=216. There are 216 possible ways to roll in this demonstration. So, the probability of rolling a sum of 9 with exactly three rolls of the die would be 25/216 and the sum of 10 with three rolls would be 27/216. Thus, the probability of obtaining a sum of 10 in exactly 3 rolls is higher.
Any comments?