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Calculating the Odds for a Yahtzee Dice Throw

 
 
JCBNLA
 
Reply Wed 18 May, 2022 12:29 am
Suppose on a piece of paper I write down 5 numbers, each number being 1, 2, 3, 4, 5, or 6. Then I shake 5 dice in a Yahtzee cup and cast them. What are the odds that the five numbers I wrote down will match the numbers on the tops of the five dice? It would not matter which die had which number, just as long as the 5 numbers written on the paper match the 5 numbers on the tops of the dice.

Is the calculation 1/6 to the power of 5 = 1/7,776? Are those odds then expressed as "1 out of 7,776"? Or does the fact that it doesn’t matter which die has which number complicate the calculation?


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engineer
 
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Reply Wed 18 May, 2022 06:40 am
@JCBNLA,
Depends on how you choose your numbers. Let's say you choose five different numbers. For the first die, the chance of getting a number on your list is 5 in 6. For the second, the chance is now 4 in six. Continuing, you can see that the probability of getting all five is 5!/6^5 = 120/7776.

But what if you duplicate numbers? If you pick all ones, there is only one solution in 7776 that meets that requirement. I think the general formula will looks like:

5!/7776/n1!/n2!/n3!/n4!/n5!/n6! where nx is the number of times you chose x as a number.

So if you choose 2,3,3,4,4, n1=0, n2=1, n3=2, n4=2 and n5 and n6=0 so you get 5!/7776/1/1/2/2/1/1 = 30/7776
JCBNLA
 
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Reply Sun 22 May, 2022 08:31 pm
@engineer,
Thank you for answering!
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