Question 1111337: Four dice are rolled and sum of numbers which appeared is noted as x.find the probability that x=13
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! The sample space for rolling 4 6-sided dice is 6^4 = 1296
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The number of ways you can get a sum of 13 is 140
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Probability (rolling a 13 with 4 6-sided dice = 140/1296 = 0.108
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The intuitive way to calculate the number of ways we can get a sum of 13 with 4 die follows
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each die has a minimum of 1 dot and a maximum of 6 dots
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force each die to have 1 dot, this leaves us 9 dots to distribute to the 4 die
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If there wasn’t an upper limit for the points to assign to the dice we’d be done. We would apply the combination with repetition formula to find all the ways
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((9C4)) = ((4+9-1)C9) = 12C9 = 12!/(9!(12-9)!) = 220 ways
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BUT, we have to account for the outlawed ways, that is, dice with a number of points ≥ 6, then
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number of allowed ways = number of unrestrained ways – number of outlawed ways
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consider 1 outlawed die with a 6, then that leaves 3 dots to distribute to the 4 die
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((4C3)) = ((4+3-1)C3) = (6C3) = 6!/(3!(6-3)!) = 20
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there are 4 die, so we need to multiply by 4, 4 * 20 = 80
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number of allowed ways = 220 - 80 = 140
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