# What is prosterior probability? Consider a scenario that a patient have liver disease is 15% probability. A test says that 5% of patients are alcholic. Among those patients diagnosed with liver disease, 7% are alcoholic. Now computer the chance of having liver disease, if the patient is alcoholic.

The posterior probability is the probability of one event occurring with some relationship to one or more other events.

Example: $$P\left ( \frac{A}{B} \right ) = \frac{P(A ∩ B)}{P(B)}$$

Where, $$P\left ( \frac{A}{B} \right )$$ is the probability of A occurring given the probability of B

Given,

Probability of patient having liver disease

P(L) = 15% = 0.15

Probability of patient being alcoholic

P(L) = 5% = 0.05

Probability of people being alcoholic given that they are having liver disease.

$$P\left ( \frac{A}{L} \right )$$ = 7% = 0.07

Now, probability of having liver disease if the patient is alcoholic: $$P\left ( \frac{L}{A} \right )$$ = ?

We know,

$$P\left ( \frac{L}{A} \right ) = \frac{P(L) . P\left ( \frac{A}{L} \right )}{P(A)}$$

= $$\frac{0.5 \times 0.07}{0.05}$$

= 0.21

= 21%

Hence, the probability of having the liver disease if the patient is alcoholic is 21%.