You toss two number cubes. if a sum of 7 or 11 comes up, you get 7 points, if not you lose 2 points. the probabilities for each of the sums is: p(2) = 1/36 p(3) = 1/18 p(4) = 1/12 p(5) = 1/9 p(6) = 5/36 p(7) = 1/6 p(8) = 5/36 p(9) = 1/9 p(10) = 1/12 p(11) = 1/18 p(12) = 1/36 the expected value of the number of points for one roll is: 0 1 3 6