Expectation vs Probability

    When a single die is rolled, the outcomes 1, 2, 3, 4, 5, & 6 all occur with equal probability. For example, the probability of a 2 is Pr(Y = 2) = 1/6. Thus, over a long series of rolls, each value will occur an equal number of times. The calculation shows that the expectation of this series is E(Y) = 3.5, which is not a value that can be obtained on any single role of the die. When a single coin is flipped, the outcomes H and T are equally probable, so  Pr(Y = H) = 1/2, and E(Y) = 0.5. Once again, "half a head" is not an observable outcome, but over the long run "50% Heads" is.

    HOMEWORK: repeat the calculations of the probability distribution and expectation for two dice.