(3) In a political district, recent polls estimate that 38% of the eligible voters prefer candidate A, 40% prefer candidate B, and the rest of the district have no preference and would not vote in the upcoming election. To win the election in the rst round a candidate needs to get majority of the votes of all eligible voters in the district and there are 1000 eligible voters. You can assume that preference of voters are independent of each other in answering the following questions. You can use R to perform the computations but be sure to show your reasoning in each case. (a) What is the probability that 400 will vote for candidate A, 450 for candidate B, and 150 will not vote in the rst round? (b) Given that candidate A has won 200 votes, what is the probabil itythat candidate B will win the election in the rst round? (0) Dene X,- as the number of people voting for candidate 2' for z' = A, B, C (i = C implies no voting). Compute the correlations between (XA,XB), (XA,X0), and (XB,X0). (3) (a) P(XA = 400, XB = 450, Xc = 150) is obtained via dmultinom(x = c(400, 450, 150), p = c(.38, .40, .22) ) = 1.014571e-10 (b) P(XB > 500| XA = 200) is given by 1 - pbinom (500, 800, 0.6451) = 0.875 (c) For pairs i, j = A, B, C, the correlations are given by Cor(Xi, Xi) - Cov(Xi, Xi) -n * Pi * Pi oxOXj n * Pi * (1 - Pi) * n * Pj * (1 -Pj) Cor(XA, XB) =-0.6392, Cor(XA, XC) = -0.4158, Cor(XB, XC) = -0.4336