1.7.3. Second example computing an expected value. Consider a group of 12 television sets, two of which have white
cords and ten which have black cords. Suppose three of them are chosen at random and shipped to a care center. What are the probabilities that zero, one, or two of the sets with white cords are shipped? What is the expected number with white cords that will be shipped? It is clear that x of the two sets with white cords and 3-x of the ten sets with black cords can be chosen in 2 x × 10 3−x ways.
The three sets can be chosen in 12 3 ways.
So we have a probability mass function as follows. f(x) = P( X = x) = 2 x 10 3−x 12 3 for x = 0 , 1 , 2 (13)
For example f(x) = P(X = x) = 2 0 10 3−0 12 3 = (1) (120) 220 = 6 11 (14)
We collect this information as in table 4. TABLE 4. Probabilities for Television Problem x 0 1 2 f(x) 6/11 9/22 1/22 F(x) 6/11