MATH/Sinus 425 INTRO. TO PROBABILITY Solutions 1. 5 couples are seated next to a round table at random- Let X be the number of couples seating next to each other. (a) Explain why there are 9! dilferont ways to seat 10 people next to a round table. (b) Let X:- be the indicator of the event where the ith couple sits next to each other. Find E [Xi] and 1E {XI-Xi] for 3' % 3' {notice these random variables are not independent). Hint: If a couple sits next to each other1 we can treat them as one \"object\" [with 2 different ways to choose the order of the seating for the couple). (0) Find E [X] and Var (X). (d) (i) Find new. Solution. (3) (b) answers: 2/9, 1f18. (c) We have E [X] z E [X1 + - - - + X5] = 5 - 7:. To nd Var (X), we compute: MED Since (E) =: 1703-1) = \"KgX, it follows that 3E[X2] = 2 - % +E[X] = Eg + 1f\" =