Suppose there are 20 rabbits 30 turtles and 40 dogs We rando

Suppose there are 20 rabbits, 30 turtles and 40 dogs. We randomly assign them into a line. Then we call a pair of adjacent animals “matching point” if they are different kinds of animals. Denote X as the number of matching points. Find E(X).

Solution

The possibilities of matching point are: Rabbit and Turtle(RT);Turtle and Dog(TD); Dog and Rabbit(DR)

The probabilities of these matching points are P(RT) = (1/20)(1/30) = 1/600

P(TD) = (1/30)(1/40) = 1/1200;P(DR) = (1/40)(1/20) = 1/800

The Values number of matching points (X) are 600(RT); 800(DR) and 1200(TD)

hence E(X) = 600*(1/600)+800*(1/800)+1200*(1/1200)=1+1+1 =3