1. in a board game, you must roll two 6-sided number cubes. you can only start the game if you roll a 3 on at least one of the number cubes.
(a) make a list of all the different possible outcomes when two number cubes are rolled.
(b) what fraction of the possible outcomes is favorable?
(c) suppose you rolled the two number cubes 100 times, would you expect at least one 3 more or less than 34 times? explain.

Answer and explanation:

There are 36 total possible outcomes with the roll of two 6-sided cubes.

{1,1} {1,2} {1,3} {1,4} {1,5} {1,6} {2,1} {2,2} {2,3} {2,4} {2,5} {2,6} {3,1} {3,2} {3,3} {3,4} {3,5} {3,6} {4,1} {4,2} {4,3} {4,4} {4,5} {4,6} {5,1} {5,2} {5,3} {5,4} {5,5} {5,6} {6,1} {6,2} {6,3} {6,4} {6,5} {6,6}

a) There are 21 DIFFERENT possible different outcomes.  For the purpose of the game, it doesn't matter if the first cube is a 3 and the second cube is a 6, so {3,6} and {6,3} are equivalent.

{1,1} {1,2} {1,3} {1,4} {1,5} {1,6} {2,2} {2,3} {2,4} {2,5} {2,6} {3,3} {3,4} {3,5} {3,6} {4,4} {4,5} {4,6} {5,5} {5,6} {6,6}

b) The question asks for the fraction of the possible outcomes (not the different possible outcomes), so we have to count how many outcomes feature a 3 and divide that by the number of possible outcomes (36).

Based on the total possible outcomes: we have 11 possibilities to get a 3, out of 36, so 11 / 36 = 30.6%

If the question was about the different possible outcomes, we would have: 6 out of 21, so 28.6%

c) A bit less, because both the total and the different possible give us a probability that is a bit less than 34%... so it's unlikely the rolls would get us 34%.