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Expert: - 6/19/2009


Question
Complete the probability distribution table for the experiment of rolling a pair of dice until a sum greater then 8 appears.

number of rolls         probability

  1
  2
  3
  4
  5
  6
  7
  8
  9
  10
  more then 10

I'm going to be quizzed on this type of probability question, my brother has worked out a formula but were not confident on the answers were coming up with.
Thanks if you can help
Sarah

Answer
There is 1 way to roll a 12, 2 ways to roll and 11, 3 ways to roll a 10, 4 ways to roll a 9 and 5 ways to roll an 8.  It can be seen that
1+2+3+4+5=15.  There are 6 ways for each die to roll and 2 dice, so the total number of rolls is 6² = 36.  This means the probability of rolling greater than an 8 is 15/36 = 5/12.

1 Roll: For this to happen in one roll is just that - 5/12.

2 Rolls: If this does not happen in the first roll, then the odds of this first roll are 7/12.  For the second roll, the probability is still 5/12, but now we have to multiply by 7/12, getting 35/144.

3 Rolls: For it not to happen in the first two rolls would by (7/12)(7/12) = 49/144.  Multiply by 5/12 and get 245/720.

4 Rolls: (5/12)(7/12)^3 = (5/12)(343/1,728) = 1,715/20,736.

5 Rolls: (5/12)(7/12)^4 = (5/12)(2401/20,736) = 12005/248,832.

6 Rolls: (5/12)(7/12)^5 = (5/12)(16807/248,832) = 84035/298,5984.

7 Rolls: (5/12)(7/12)^5 = (5/12)(117,649/2,985,984) =
588,245/35,831,808

8 Rolls: (5/12)(7/12)^6 = (5/12)(823,543/35,831,808) =
40,117,715/429,981,696.

Anyway, that's how to do it.

The results are
1   0.416666667
2   0.243055556
3   0.141782407
4   0.082706404
5   0.048245403
6   0.028143151
7   0.016416838
8   0.009576489
9   0.005586285
10   0.003258666

In scientific notation to two decimals, they are
1   4.17E-01
2   2.43E-01
3   1.42E-01
4   8.27E-02
5   4.82E-02
6   2.81E-02
7   1.64E-02
8   9.58E-03
9   5.59E-03
10   3.26E-03

Oh yeah, and the chance of going over 10 is 1.58E-04.

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