Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Example: We have 2 boxes. The first box contains 2 green and 7 red balls. The
second box contains 4 green and 3 red balls. We select a box at random, then a
ball at random. If we picked a red ball, what is the probability that it came from
the first box?
• Let E be the event that we chose a red ball. Thus, E is the event that we
chose a green ball. Let F be the event that we chose a ball from the first box.
Thus, F is the event that we chose a ball from the second box. p(F) = p(F) =
0.5 since we pick a box at random.
• We want to calculate p(F|E) = p(E3F) / p(E), which we will do in stages.
• p(E|F) = 7/9 since there are 7 red balls out of 9 total in box 1. p(E|F) = 3/7
since there are 3 red balls out of a total of 7 in box 2.
• p(E3F) = p(E|F)p(F) = 7/18 = 0.389 and p(E3F) = p(E|F)p(F) = 3/14.
• We need to find p(E). We do this by observing that E = (E3F)2(E3F),
where E3F and E3F are disjoint sets. So, p(E) = p(E3F)+p(E3F) = 0.603.
• p(F|E) = p(E3F) / p(E) = 0.389 / 0.603 = 0.645, which is greater than the 0.5
from the second bullet above. We have improved our estimate!
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Example: Suppose one person in 100,000 has a particular rare disease and that
there is an accurate diagnostic test for this disease. The test is 99% accurate when
given to someone with the disease and is 99.5% accurate when given to someone
who does not have the disease. We can calculate
(a) the probability that someone who tests positive has the disease, and
(b) the probability that someone who tests negative does not have the disease.
Let F be the event that a person has the disease and let F be the event that this
person tests positive. We will use Bayes theorem to calculate (a) and (b), so have
to calculate p(F), p(F), p(E|F), and p(E|F).
• p(F) = 1 / 100000 = 10 45 and p(F) = 1 4 p(F) = 0.99999.
• p(E|F) = 0.99 since someone who has the disease tests positive 99% of the
time. Similarly, we know that a false negative is p(E|F) = 0.01. Further,
p(E|F) = 0.995 since the test is 99.5% accurate for someone who does not
have the disease.
• p(E|F) = 0.005, which is the probability of a false negative (100 4 99.5%).
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