How much better than random chance? I want to demonstrate to potential customers that I can select certain people with far greater probability than random chance. I ask customers to allow me to assess 100 people. I further ask that they include in that 100 about 20 people with a specific incident in their history. If I then identify 22 people I believe are in the special group, and 18 of them are actual matches to the special group, how much better than random chance did I do?
 A: You did not specify how many special group members were actually included among the 100 people assessed, but since you asked for about 20 such people to be included, let's assume there were exactly 20.
If you chose 22 people at random, the chances of getting at least 18 correct would be: $$ {{20 \choose 18}{80 \choose 4}+{20 \choose 19}{80 \choose 3}+{20 \choose 20}{80 \choose 2}\over {100 \choose 22}}=4.121*10^{-14}$$ which is less than 1 in 24 trillion.  In order words, you are clearly better than random.
It is important to note though that this is only true if the customers indeed only included about 20 special members. If however, in an attempt to make you look good they included say 90 special members, then your selections very well could be random.
Response to the OP's comment. The "stacked values in parentheses" count subsets. For example, the ${100 \choose 22}$ in the denominator is the number of ways to choose a group of 22 people from among the 100. The first term in the numerator counts the ways to choose (correctly) exactly 18 with the incident from among the 20 times the number of ways to choose 4 (incorrectly) from the other 80. For a correct score of 18 or better you should add the ways to get 19 right and 3 wrong (the second term) and 20 right, 2 wrong (the last term). You can find the formulas for these counts at the wikipedia page for binomial coefficients. 
A: Consider this table
             incident   no incident     total

selected        18          4             22

not selected     2         76             78

total           20         80            100

Then your false positive rate is $4/22 \approx 18\%$ while your false negative rate is $2/20 = 10\%$.
If you selected at random (each person with a 20% probability) the table would be something like
             incident   no incident     total

selected         4         16             20

not selected    16         64             80

total           20         80            100

so the false positive and false negative rates would both be about $80\%$.
You're doing a lot better than random.
