The probability of selecting no bad ones is the probability of
selecting 2 from the 8 good bulbs, which is C(8,2)/C(10,2) or as
some people write it (8C2)/(10C2) = 28/45
The probability of selecting 1 good one and one bad one, is the
probability of selecting 1 from the 8 good bulbs, and one from
the 2 bad ones, which is [C(8,1)C(2,1)]/C(10,2) or as some people
write it (8C1*2C1)/(10C2) = 16/45.
The probability of selecting two bad one, is C(2,2)/C(10,2) or
as some people write it (2C2)/(10C2) = 1/45.
The probability distribution function is:
x p(x)
0 28/45
1 16/45
2 1/45
Notice that the sum of those three probabilities is
28/45 + 16/45 + 1/45 = 45/45 = 1.
The expectation is
E(x) = 0*(28/45) + 1*(16/45) + 2*(1/45) = 28/45 + 16/45 + 2/45 =
46/45 = 1.022222
If we were to reach into the same box and draw out 2 bulbs
thousands of times we would expect to average picking out
1.02222 bad bulbs. It doesn't mean that we would ever expect
to draw out a fraction of a bulb! That would be impossible.
We just mean that we would expect to average 1.02222 bulbs
if we were to repeat the experiment many times.
Edwin
