Casting Dice for Penance

from Freund's Mathematical Statistics

Counting the number of outcomes in games of chance has been a popular pastime for many centuries. This was of interest not only because of the gambling that was involved, but also because the outcomes of games of chance were often interpreted as divine intent. Thus, it was just about 1000 years ago that a bishop in what is now Belgium determined that there are 56 different ways in which three dice can fall provided one is interested only in the overall result and not which die does what. This bishop assigned a virtue to each of these possibilities and after a parishoner went to confession he was required to meditate for some time on the virtue that corresponded to his cast of the dice.

Was the bishop correct by saying there were 56 possibilities?

Determine the size of the sample space.

Solution:

Let A be the event that all three dice have the same number of points. Then N(A) is the number of ways in which the three dice can all come up with the same number of points.

N(A) = 6

Let B be the event that two of the dice show the same number of points. Then N(B) is the number of ways that exactly two of the dice can come up with the same numbers.

N(B) = 30

Let C be the event that none of the dice show the same number of points. N(C) is the number of ways that this can happen.

N(C) = 20

Since A, B and C are mutually exclusive

N(AÇBÇC) = N(A) + N(B) + N(C)