A music device stores a playlist of 6 song clips, among which songs 1, 2 and 3 are sung by
singer A, songs 4 and 5 by singer B, and song 6 by singer C. Suppose that the device plays
the 6 songs according to a random playing order, which is equally likely to be any one of the
6!= 720 permutations of the 6 songs.
Define E to be the event that song 6 is played first and S to be the event that no two successive
songs are sung by the same singer. For i, je {1,..., 6}, define Ei, to be the event that song i
is followed immediately by song j.
(a) Show that Pr(S|E) = 1/10.
(b) Show that Pr(E12) = 1/6.
(c) (i) Calculate Pr(E12E23).
(ii) Calculate Pr(E12E45).
(d) (i) Calculate Pr(E12 E23 E31).
(ii) Calculate Pr(E12 E23 E45).
(e) Express S in terms of the events E, (i, je {1,..., 6}).
(f) Use inclusion-exclusion principle to calculate Pr(S).
(g) Using (a) and (f), calculate the conditional probability Pr(EIS).