Problem 1

Which of the following outcomes has the highest probability? Why?

1) The outcome has three 1's and three 6's.

There are 6 !/(3 ! 3 !) ways to order three 1 ' s and three 6 ' s and each roll has a probability of 1/6so the answer is :

P[1] = 6 !/(3 ! 3 !) (1/6)^6 //N

0.000428669

2) The outcome has the sequence {1,2,3,4,5,6}.

P[1, 2, 3, 4, 5, 6] = (1/6)^6 //N

0.0000214335

3) The sum of the results of the six rolls is greater than 19, given that the  sum of the first five rolls is 15.

P[19 | 15] = P[5or 6] = 1/6 + 1/6 = 1/3

Created by Mathematica  (October 8, 2003)