Question

A playlist contains 10 rock songs, 3 country songs, 5 R&B songs, and 2 blues songs. In shuffle mode, each song is played exactly once, and all possible equal orderings are equally likely. Suppose that a person starts this playlist in shuffle mode and continues until a country music song plays, and then stops. Let X denote the number of songs played before the country music song (but not including the country music song itself). [[Hint: Write X=X1+?+X17, where Xj=1 if the jth non-country song is played before all of the country songs, or Xj=0 otherwise.]]

a. Find E(X).

b. Find Var?(X).

Answer 1

a. Find E(X)

We have E(X) = E(X₁ + · · · + X₁₇)

= E(X₁) + · · · + E(X₁₇).

Also E(Xj ) = 1/4,

so it follows that E(X) = (17)*(1/4)

= 17/4

E(X) = 4.25

b. Find Var(X)

We have E(X² ) = E((X₁ + · · · + X₁₇) ² ), which has 17 terms of the form E(X_j² )

and 17² ? 17 = 272 terms of the form E(X_iX_j ).

Also E(X_j² ) = E(Xj ) = 1/4

and E(X_iX_j ) = (2)*(1/5)*(1/4) = 1/10

Thus E(X² ) = (17)*(1/4) + (272)*(1/10) = 31.45

So altogether we have Var(X) = E(X² ) ? (E(X))²

= 31.45 ? (4.25)²

Var(X) = 13.3875.