We have two fair three-sided dice, indexed by =1,2 . each die has sides labelled 1 , 2 , and 3 . we roll the two dice independently, one roll for each die. for =1,2 , let the random variable represent the result of the th die, so that is uniformly distributed over the set {1,2,3} . define =− . calculate the numerical values of following probabilities, as well as the expected value and variance of : (a) p(x = 0) = (b) p(x = 1) = (c) p(x = −2) = (d) p(x = 3) = 2. let y = x2 . calculate the following probabilities: (a) p(y = 0) = (b) p(y = 1) = (c) p(y = 2) =

Ily begins solving the equation 4(x – 1) – x = 3(x + 5) – 11. her work is shown below. 4(x – 1) – x = 3(x + 5) – 11 4x – 4 – x = 3x + 15 – 11 3x – 4 = 3x + 4 how can her partial solution be interpreted?

At the olympic games, many events have several rounds of competition. one of these events is the men's 100-meter backstroke. the upper dot plot shows the times (in seconds) of the top 8 finishers in the final round of the 2012 olympics. the lower dot plot shows the times of the same 8 swimmers, but in the semifinal round. which pieces of information can be gathered from these dot plots? (remember that lower swim times are faster.) choose all answers that apply: a: the swimmers had faster times on average in the final round. b: the times in the final round vary noticeably more than the times in the semifinal round. c: none of the above.

Does the function satisfy the hypotheses of the mean value theorem on the given interval? f(x) = 4x^2 + 3x + 4, [−1, 1] no, f is continuous on [−1, 1] but not differentiable on (−1, 1). no, f is not continuous on [−1, 1]. yes, f is continuous on [−1, 1] and differentiable on (−1, 1) since polynomials are continuous and differentiable on . there is not enough information to verify if this function satisfies the mean value theorem. yes, it does not matter if f is continuous or differentiable; every function satisfies the mean value theorem.