Applied statistics/Tutorials

Rules of chance

The addition rule

For two mutually exclusive events, A and B,
the probability that either A or B will occur is equal to the probability that A will occur plus the probability that B will occur,

P(A or B) = P(A) + P(B).

The multiplication rule

For two independent (unrelated) events, A and B,
the probability that A and B will both occur is equal to the probability that A will occur multiplied by the probability that B will occur,

P(A and B) = P(A) x P(B)

Bayes' theorem

The probability that event A will occur, given that event B has occurred is equal to the probability that B will occur, given that A has occurred, mutiplied by the probability that A will occur divided by the probability that B will occur,

P(A/B) = P(B/A) x P(A)/P(B).

The false positive question

The question:
If a test of a disease that has a prevalence rate of 1 in 1000 has a false positive rate of 5%, what is the chance that a person who has positive result actually has the disease.
The answer:
2%
Proof:
Let A denote the event of having the disease and, B the event of having been tested positive (for the purpose of applying Bayes'theorem). Then P(B/A) which is the probability of having been tested positive when having the disease, can be taken asequal to 1;
And P(A) is the probability of having the disease, which with a prevalence of 1 in 1000 must be equal to 1/1000<
And P(B) is the probability of being tested positive, which can be arrived at by 3 steps:
Step 1 is to observe that since the prevalence of the disease is 1 in 1000, 999 persons out of every 1000 are healthy. Step 2 is to recall that for each healthy person the probability of being tested positive is 5% or 1 in 20. Step 3 is to apply the multiplication rule and get the answer:

P(B) = 999/1000 multiplied by 1/20 or, near enough 1/20.

So applying Bayes' theorem, the probability of having the disease, given that you have been tested positive is given by

P(A/B) = P(B/A) x P(A)/P(B), or:

   =     1    x  (1/1000)/(1/20) - which is 0.02, or 2%.