## Bayes'
Theorem

** Bayes' theorem** is
stated mathematically as:

$P(A\mid B)={\frac {P(B\mid A)\,P(A)}{P(B)}},$
where $A$ and $B$ are **events** and $P(B)\neq 0$.

$P(A\mid B)$ is a

**conditional probability**:
the

*likelihood *of event

$A$ occurring, given that

$B$ is true.

$P(B\mid A)$ is
also a

**conditional probability**: the

*likelihood *of
event

$B$ occurring, given that

$A$ is true.

$P(A)$ and

$P(B)$ are the

**marginal
probabilities** of observing

$A$ and

$B$, independently of each
other.

### Example: Drug
testing

Suppose a blood test used to
detect the presence of a particular banned drug is **99% ****sensitive**
and **99% ****specific**.
That is, the test will produce 99% **true positive** results
for drug users and 99% **true negative** results for
non-drug users. Suppose that **0.5%** of people are users of
the drug. What is the **probability** that *a randomly
selected individual who ***tests positive is a user**?

${\begin{aligned}P({\text{User}}\mid
{\text{+}})&={\frac {P({\text{+}}\mid
{\text{User}})P({\text{User}})}{P(+)}}\\&={\frac
{P({\text{+}}\mid
{\text{User}})P({\text{User}})}{P({\text{+}}\mid
{\text{User}})P({\text{User}})+P({\text{+}}\mid
{\text{Non-user}})P({\text{Non-user}})}}\\[8pt]&={\frac
{0.99\times 0.005}{0.99\times 0.005+0.01\times
0.995}}\\[8pt]&\approx 33.2\%\end{aligned}}$
** Even if an individual tests
positive, it is more likely than not (1 - 33.2% = 66.8%) that
s/he does ***not* use the drug. Why? Even though the
test appears to be highly accurate, *the number of non-users
is *very large *compared to the number of users*.
Then, the count of *false positives* will outweigh the
count of *true positives*.

To see this with actual
numbers, if 1,000 individuals are tested, we expect 995
non-users and 5 users. Among the 995 non-users,**
0.01 × 995 ≃ 10 false positives** are
expected. Among the 5 users, **
0.99 × 5 ≈ 5 true positives** are
expected. Out of 15 positive results, only 5, ~33%, are genuine.

The importance of **specificity**
in this example can be seen by calculating that even if *
sensitivity *is improved to 100%, but *specificity *remains
at 99%, then the **probability that a person who tests
positive is a drug user only rises from 33.2% to 33.4%**.
Alternatively, if *sensitivity *remains 99%, but
* specificity *is improved to 99.5%, then the **probability
that a person who tests positive is a drug user rises to about
49.9%.**