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Dice Probability
- Thread starter kenziejwu
- Start date
One of the easiest ways may be to just list out the 36 outcomes from rolling 2 dice and count how many fit the respective criteria.
We can also use generating functions.
\(\displaystyle \left(\sum_{k=1}^{6}x^{k}\right)^{2}\)
This can be expanded out to \(\displaystyle x^{2}+2x^{3}+3x^{4}+4x^{5}+5x^{6}+6x^{7}+5x^{8}+4x^{9}+3x^{10}+2x^{11}+x^{12}\) Then, look at the coefficients that correspond to the exponent that represents the roll.
For instance, how many ways to roll a sum of 6?. Look up x^6 in the expansion. Its coefficient is the number of ways there is a sum of 6.
As we can see, there are 5 ways to get a sum of 6 and 4 ways to get a sum of 9. So, the probability of summing 6 or 9 is 9/36=1/4.
Or make a table:
The sum of two dice:
\(\displaystyle \begin{array}{c|c|c|c|c|c|c} \;\ &1&2&3&4&5&6\\ \hline 1&2&3&4&5&6&7\\ 2&3&4&5&6&7&8\\ 3&4&5&6&7&8&9\\ 4&5&6&7&8&9&10\\ 5&6&7&8&9&10&11\\ 6&7&8&9&10&11&12\end{array}\)
Count up how many are in the table and divide by 36. There are 5 6's in the table.
We can also use generating functions.
\(\displaystyle \left(\sum_{k=1}^{6}x^{k}\right)^{2}\)
This can be expanded out to \(\displaystyle x^{2}+2x^{3}+3x^{4}+4x^{5}+5x^{6}+6x^{7}+5x^{8}+4x^{9}+3x^{10}+2x^{11}+x^{12}\) Then, look at the coefficients that correspond to the exponent that represents the roll.
For instance, how many ways to roll a sum of 6?. Look up x^6 in the expansion. Its coefficient is the number of ways there is a sum of 6.
As we can see, there are 5 ways to get a sum of 6 and 4 ways to get a sum of 9. So, the probability of summing 6 or 9 is 9/36=1/4.
Or make a table:
The sum of two dice:
\(\displaystyle \begin{array}{c|c|c|c|c|c|c} \;\ &1&2&3&4&5&6\\ \hline 1&2&3&4&5&6&7\\ 2&3&4&5&6&7&8\\ 3&4&5&6&7&8&9\\ 4&5&6&7&8&9&10\\ 5&6&7&8&9&10&11\\ 6&7&8&9&10&11&12\end{array}\)
Count up how many are in the table and divide by 36. There are 5 6's in the table.
D
Deleted member 4993
Guest
kenziejwu said:When two dice are rolled find the probability of getting each of the following:
1. a sum of 6 or 9
2. a sum less than 3 or greater than 7
3. a sum that is divisible by 4
4. a sum less than 11
You can 36( = 6[sup:1s17h9k8]2[/sup:1s17h9k8]) positions when you throw two dice.
How many of those add upto 6?
How many of those add upto 9?
How many of those add upto 6 or 9?
Now continue....
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