Probabilities when Rolling 10 Dice

Date: 07/29/99 at 19:25:41
From: Jeffrey
Subject: A question of probability and odds.

The house rolls 10 dice, one die at a time. What are the odds of the 
following:

1. A player rolling the same numbers in the same order as the house.

2. A player rolling the same numbers as the house without regard to 
   the order in which they were rolled.

3. A player rolling 9 of 10 numbers the same as the house without 
   regard to the order in which they were rolled.

4. A player rolling 8 of 10 numbers the same as the house without 
   regard to the order in which they were rolled.

Assumptions: 10 standard six-sided dice numbered 1 to 6.

Thank you in advance.

Date: 08/01/99 at 06:27:12
From: Doctor Anthony
Subject: Re: A question of probability and odds.

1. A player rolling the same numbers in the same order as the house.

Each of the 10 dice could produce 6 different numbers, so there are 
6^10 different arrangements.

The probability that a player will get the same numbers in the same 
order is:

     1/6^10

2. A player rolling the same numbers as the house without regard to 
   the order in which they were rolled.

This is a more difficult calculation. We shall approach it by 
considering that we have 6 cells, representing the possible number on 
the face of any die, and that 10 balls are to be distributed into the 
6 cells. The number of balls in any cell will represent the number of 
times that the cell number occurred when the 10 dice are rolled. It is 
possible of course that some cells (from 0 to 5) could be empty.

We can represent the situation graphically as follows:

     |***| |**|****| |*|   representing occupancy 3, 0, 2, 4, 0, 1

in the 6 cells. That is 3 1's, 0 2's, 2 3's, 4 4's, 0 5's and 1 6.

With 6 cells we must start and end with a |, but there are five |'s 
and 10 *'s to permute in every possible way. That is 15 objects, 5 
being alike of one kind and 10 being alike of a second kind. The 
number of arrangements is:

      15!
     ------  = 3003
     5! 10!

So now the probability that the player gets the same distribution of 
numbers as the house is:

     1/3003

3. A player rolling 9 of 10 numbers the same as the house without 
   regard to the order in which they were rolled.

There are 3003 ways in which the 10 balls could be placed in the 
cells. Imagine that all 10 are correct, that is, we have the same 
distribution of balls in the cells for the house and the player. We 
must now think of the number of ways that we could end up with just 9 
correct. This can be achieved by taking one ball from one cell and 
transferring it to another cell. There will then be a shortage of 
balls in one cell but a surplus in another cell. So we must count the 
number of ways that we could make the transfer.

Consider first the case with no empty cells.

With no cells empty we can represent the situation as in the diagram 
below:

     |**|*|***|**|*|*|   with occupancy  2, 1, 3, 2, 1, 1

Now we start and end with a |, but we cannot have any of the other 5 
|'s from being together. We must choose 5 of the 9 gaps between the 10 
*'s. This can be done in C(9,5) = 126 ways.

If there are no empty cells we could take the ball in 6 different ways 
and place it in another cell in 5 ways - giving 30 different 
arrangements with one ball out of place. The probability of having one 
ball out of place in the case of no empty cells in the original house 
arrangement is:

     30/126 = 5/21  

However this must be multiplied by the probability that there are no 
empty cells. This is given by 126/3003.

So total probability of having no empty cell and having 9 out of 10 
correct is:

     126/3003 x 30/126  =  30/3003

We can see that the number 126 has cancelled out, so we do not need to 
calculate the number of ways that we could have exactly, 0,1,2,...,5 
empty cells. All we require is the number of ways that the transfers 
between cells can occur for the various configurations and put this 
total over 3003.

If one cell is empty then the number of ways of making the transfer is 
calculated as follows.

We can choose the cell from which to take a ball in 5 ways and place 
it in another cell also in 5 ways. (We can now use the empty cell into 
which to move the ball as one option). This gives 5 x 5 = 25 ways of 
having 9 out of 10 correct with one cell empty. The required 
probability is:

     25/3003

With two cells empty the cell to take the ball from can be chosen in 
4 ways, and the cell into which to place the ball in 5 ways, giving 
4 x 5 = 20 ways of moving one ball. The probability that 9 out of 10 
are correctly placed when two cells are originally empty is then:

     20/3003

With three cells empty we can take one ball in 3 ways and place it in 
5 ways = 15 ways.

Required probability with 9 out of 10 correct and 3 cells empty is:

     15/3003

With 4 cells empty we can take one ball in 2 ways and place it in 
5 ways = 10 ways.

Required probability that 9 out of 10 correct and 4 cells empty is

     10/3003

With 5 cells empty we can take one ball in 1 way and place it in 
5 ways = 5 ways.

Required probability that 9 out of 10 correct and 5 cells empty is:

     5/3003

So the total probability that 9 out of 10 are correct is:

         30 + 25 + 20 + 15 + 10 + 5     105       5
     =  ---------------------------  =  ----  =  ---
                   3003                 3003     143


4. A player rolling 8 of 10 numbers the same as the house without 
   regard to the order in which they were rolled.

I will leave this for you to puzzle over.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/