Bayes' Theorem

Bayes' Theorem allows us to calculate the probability of a specific event A, based on the information we have about event B. $$ p(A|B) = \frac{ p(A) \cdot p(B|A) }{ p(B) } $$

p(A|B) and p(B|A) are conditional (posterior) probabilities
p(A) and p(B) are prior (simple) probabilities

How does conditional probability work? Conditional probability p(A|B) represents the likelihood of event A occurring, given that event B has already occurred.

A practical example

A population is composed of 40% males and 60% females.

All men wear pants, while women wear pants 50% of the time and skirts the other 50% of the time.

statistical population

An observer sees someone in the distance wearing pants.

What is the probability that this person is female?

The prior probability ( event A ) of the person being female is 60%.

$$ p(A) = 0.6 $$

Note. This is a straightforward estimate based on the population makeup, but it’s not precise enough to answer the question reliably.
statistical population

The prior probability that a person is wearing pants ( event B ) is 70%.

$$ p(B) = 0.7 $$

Note. All men wear pants (40% of the population), and half of the women do (30%). Therefore, 70% of the population wears pants.
explanation

According to the data, the conditional probability (posterior) that a woman (event A) is wearing pants (event B) is 50%.

$$ p(B|A) = 0.5 $$

Note. Half of the women in the population wear pants. This 50% refers only to the female population, not the total population.
50% of the female population wears pants

Now, we can apply Bayes' Theorem to calculate the probability that the person wearing pants (event B) is female (event A).

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

By substituting the known probabilities, we can solve the problem.

$$ p(A|B) = \frac{ 0.6 \cdot 0.5 }{ 0.7 } $$

$$ p(A|B) = 0.43 $$

So, there is a 43% chance that the person observed from a distance is female.

This is the conditional probability of event A (female), given the information we have about event B (pants).

Note. Initially, the prior probability p(A) of the person being female was 60%. The conditional probability, however, is reduced to 43%, which is more accurate because it incorporates the additional information from event B.

This is a simple example of how Bayes' Theorem can be applied in practice.

And so on.