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Dice Odds Proposition Bet

It's very easy to be fooled by dice odds and what may seem to be an attractive, uncomplicated proposition game with dice may not be quite as simple or as good as it first seems.

Someone who knows a little about dice odds will know that with two dice there are more ways to make a 7 than either a 6 or 8.  Right?  6 ways to make a 7 and only 5 ways to make a 5 or 8.  Now this is absolutely true but think about the following proposition bet.

  • I'll bet that with two dice I can roll an 8 and a 6 before you roll two 7s.

Now at first this seems like a good bet for the person offered this proposition.  After all there really are more ways to make a 7 than an 8 or 6.  So you would think that it is perfectly reasonable and logical to assume that two 7s would come up first before both an 8 and a 6.  Wrong!  Take a look at the analysis below.

Remember that only the numbers 6, 8 and 7 have any significance in this bet.  All the other numbers are meaningless and are just simply ignored.  Look at the following table illustrating all the possible combinations of two dice.

                       Winning combinations.  Losing combinations

d1/2 1 2 3 4 5 6
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

The proposition offered doesn't specify the order in which the 8 and 6 are made.  If we count the total number of possible combinations of 8 and 6 and compare it to the total number of possible combinations of 7 we can see that there are a total of 10 ways to make an 8 or a 6 and only 6 ways to make a 7.  So with every single roll of the dice we are more likely to get either an 8 or a 6 than get a 7.

The odds are in favour of rolling an 8 and a 6 before two 7s, 55.55% of the time.



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