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.

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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%.

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