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Expert: - 4/16/2011


Question
In the game of craps. What is the probability of both the 5 and the 9 appearing before the 7?

Answer
A 5 can appear as a 1-4, 2-3, 3-2, or 4-1, so there are 4 ways.
A 9 can appear as a 3-6, 4-5, 5-4, or 6-3, so there are also 4 ways.
Since 4+4 = 9, there are 8 ways for a 5 or 9 to appear.

A 7 can appear as a 1-6, 2-5, 3-4, 4-3, 5-2, or 6-1, so there are 6 ways for a 7 to appear.

That means there is a 8/6 chance of rolliing a 5 or a nine first.

That answers your quesiton, but as for the game of craps, read on ...

There are 36 ways for two six sided dice to appear.
This means the chance of a 5 or 9 is 8/36 and the chance of a 7 is 6/36.
To reduce with a common denominator, that is 4/18 for a 5 or 9 and 3/18 for a 7.

This means there there is a 4/18 chance of a 5 or 9, a 3/18 chance of a 7,
and a 11/18 chance of , rolling again.

When another roll is done, it is still the same chance.
To figure it out, that is 7 rolls that end the game, and 4 are a 5 or 9, 3 are a 7.

From what I looked up at http://en.wikipedia.org/wiki/Craps , and here's what I get:
Any number of players can play with one person working for the casino (the dealer).
Players may have an option to bet on what outcome they expect.
The person throwing the dice is the "shooter".
The dealer has to be, and all of the other players have an option to bet.


Whne shooter, each player bets on "Pass" or "Don't Pass".
"Pass/Don’t pass"  is sometimes called "Win/Don't win" or "Right/Wrong".
The shooter will lose if a 7 is rollled, and the new shooter is the next person on the right.
Sometimes five dice are used, and the shooter needs to choose two,
 then return the other three to the dealer.
A 1st roll of 2, 3, or 12 is "craps", or a loss (chances are 1/26,2/26,1/36; total = 4/36).
A 1st roll of 7 or 11 is a win (chances are 6/26,2/36; total = 8/36.
A roll of 4, 5, 6, 8, 9, or 10 determines what the shooter needs to roll again before a 7.

Now on the first row, it turns out to be a 1/3 chance of stopping right there.
The chances of each number are
2   0.027777778
3   0.055555556
4   0.083333333
5   0.111111111
6   0.138888889
7   0.166666667
8   0.138888889
9   0.111111111
10   0.083333333
11   0.055555556
12   0.027777778
  
On the first roll, there is a 1/9 chance of losing and a 2/9 chance of winning.
On all of the other rolls, the chances are you will lose.

For a 4, there is a 3/36 chance of rolling it again, but a 6/36 chance of a 7.
That means 1/3 of the time you win and 2/3 of the time you lose.

For a 5, there is a 4/36 chance of winning, so it turns out to be 2/5 win, 3/5 loss.

For a 6, there is a 5/36 chance of winning, so it turns out to be 5/11 win, 6/11 loss.

An 8 is the same as a 6, a 9 is the same as a 5, and a 10 is the same as a 4.

When these are all added together, the net winning is -0.014141414
which is a loss of 7/495 times whatever is bet.

This is a little less than a penny and a half loss on each dollar bet per roll.

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