Consider a group ofn4 people, numbered from1ton. For each pair(i,j) withij, personi and personjare friends, with probabilityp. Friendships are independent for different pairs. Thesen people are seated around a round table. For convenience, assume that the chairs are numbered from1ton, clockwise, withn located next to 1, and that personi seated in chairi. In particular, person1and personn are seated next to each other. If a person is friends with both people sitting next to him/her, we say this person ishappy. LetH be the total number of happy people.

We will findE[H]andVar(H)by carrying out a sequence of steps. Express your answers below in terms ofp

and/orn.

LetIibe a random variable indicating whether the person seated in chairiishappyor not (i.e.,Ii=1if personiis happy andIi=0otherwise). FindE[Ii] Fori=1,2,...,n,

E[Ii]=

FindE[H]

(Note: The notationaE[H] means thata is defined to beE[H] The simpler variable names will be used in the last question of this problem.)

aE[H]=

SinceI1,I2,...,Inare not independent, the variance calculation is more involved. For anyk{1,2,...,n}, findE[Ik2]

bE[Ik2]=

For anyi{1,2,...,n}, and under the conventionIn+1=I1, findE[IiIi+1]

cE[IiIi+1]=

Suppose thatij and that personsiandj are not seated next to each other. FindE[IiIj]

dE[IiIj]=

Give an expression forVar(H), in terms ofn, and the quantitiesa,b,c,d defined in earlier parts.

Var(H)=