Let X1, Y1, X2, Y2, . . . be independent random variables, each uniformly distributed in the unit interval [0, 1], and let W = (X1 + . . . + X500) − (Y1 + . . . + Y500) 500 . Let, 500 is a big number. Using the Central Limit Theorem (CLT), find a good approximation to the probability P(|W − E[W]| < 0.01).

Exercise: counting committees 0.0/2.0 puntos (calificable) we start with a pool of n people. a chaired committee consists of k≥1 members, out of whom one member is designated as the chairperson. the expression k(nk) can be interpreted as the number of possible chaired committees with k members. this is because we have (nk) choices for the k members, and once the members are chosen, there are then k choices for the chairperson. thus, c=∑k=1nk(nk) is the total number of possible chaired committees of any size. find the value of c (as a function of n ) by thinking about a different way of forming a chaired committee: first choose the chairperson, then choose the other members of the committee. the answer is of the form c=(α+nβ)2γn+δ. what are the values of α , β , γ , and δ ?