PX284 - A4 - bayes' theorem

conditional probability

the probability of $A$ happening given that $B$ has happened is written as: $P (A ∣ B)$

joint probability:

the probability that $A$ and $B$ both occur if they are independent:

P (A \cap B) = P (A) \cdot P (B)

if they are not independent:

P (A \cap B) = P (A ∣ B) \cdot P (B) = P (A ∣ B) \cdot P (A)

bayes' theorem

Formula

P (A ∣ B) = \frac{P (B ∣ A) \cdot P (A)}{P (B)}

example

an athlete fails a drugs test which is $95 %$ accurate. what is the likelihood that they are guilty if on average, only $1 %$ of athletes take drugs
here, $P (D) = 0.01$ , so, $P (\bar{D}) = 0.99$
$P (+ v e ∣ D) = 0.95$ ,
$P (+ v e ∣ \bar{D}) = 0.05$
$P (- v e ∣ \bar{D}) = 0.95$
$P (- v e ∣ D) = 0.05$
using bayes' theorem:

P (D ∣ + v e) = \frac{P (+ v e ∣ D) P (D)}{P (+ v e)}

P (+ v e) = P (+ v e ∣ D) \cdot P (D) + P (+ v e ∣ \bar{D}) \cdot P (\bar{D}) = 0.06

P (D ∣ + v e) = \frac{0.95 \times 0.01}{0.06} = 0.16