During the course of a day a machine turns out two items, one in the morning and one in the afternoon. The quality of each item is measured as good (G), mediocre (M), or bad (B). The long-run fraction of good items the machine produces is 0.4, the fraction of mediocre items is 0.2, and the fraction of bad items is 0.4.

The machine is producing 2 items each day and each item can be good (G), mediocre (M) or bad (B). The probability of producing these products is given as:

P(G) = 0.4

P(M) = 0.2

P(B) = 0.4

The sample space (quality of each of the two items) for the experiment can be written as:

GG, GM, GB, MG, MM, MB, BG, BM, BB

To calculate the probability of each of the sample points mentioned above, simply multiply the probability of each product type.

P(GG) = P(G) x P(G) = 0.4 x 0.4 = 0.16

P(GM) = P(G) x P(M) = 0.4 x 0.2 = 0.08

P(GB) = P(G) x P(B) = 0.4 x 0.4 = 0.16

P(MG) = 0.2 x 0.4 = 0.08

P(MM) = 0.2 x 0.2 = 0.04

P(MB) = 0.2 x 0.4 = 0.08

P(BG) = 0.4 x 0.4 = 0.16

P(BM) = 0.4 x 0.2 = 0.08

P(BB) = 0.4 x 0.4 = 0.16

A good items returns a profit of $2

A mediocre item returns a profit of $1

A bad item returns a profit of $0

To calculate profit for each sample point:

Sample point Profit($)

GG 2+2 =4

GM 2+1=3

GB 2+0=2

MG 1+2=3

MM 1+1=2

MB 1+0=1

BG 0+2=2

BM 0+1=1

BB 0+0=0

Total profit of the day can be calculated by multiplying the probability of each sample point with its profit for the day.

E(X) = P(X).X

Total Profit = (0.16)(4) + (0.08)(3) + (0.16)(2) + (0.08)(3) + (0.04)(2) + (0.08)(1) + (0.16)(2) + (0.08)(1) + (0.16)(0)

= 0.64 + 0.24 + 0.32 + 0.24 + 0.08 + 0.32 + 0.08 + 0

Expected Value of Total Profit E(X) = $1.92