Explain Bayes’ Theorem with an example.

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Bayes’ Theorem is a mathematical formula used to update the probability of a hypothesis based on new evidence. It combines prior knowledge with observed data to make more accurate predictions.

The Formula:

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

Where:

• P(A|B) = Probability of A given B (posterior)

• P(B|A) = Probability of B given A (likelihood)

• P(A) = Probability of A (prior)

• P(B) = Probability of B (evidence)

Example: Medical Test for a Disease

Suppose:

• 1% of the population has a rare disease → P(Disease) = 0.01

• The test is 99% accurate:

  • If someone has the disease: P(Positive | Disease) = 0.99

  • If someone does not have the disease: P(Positive | No Disease) = 0.01

You take the test and get a positive result. What’s the chance you actually have the disease?
We want P(Disease | Positive).

Apply Bayes’ Theorem:

$$P(Disease|Positive) = \frac{P(Positive|Disease) \cdot P(Disease)}{P(Positive)}$$

First, calculate P(Positive) (total chance of a positive test):

$$P(Positive) = (0.99 \cdot 0.01) + (0.01 \cdot 0.99) = 0.0099 + 0.0099 = 0.0198$$

Now plug into Bayes’ formula:

$$P(Disease|Positive) = \frac{0.99 \cdot 0.01}{0.0198} ≈ 0.5$$

Result:

Even with a positive test, there’s only a 50% chance you actually have the disease—because the disease is so rare.

Summary:

Bayes’ Theorem helps you update beliefs (probabilities) based on new evidence, especially when dealing with uncertainty or rare events.

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