Bayes' Theorem
Bayes' theorem, named after the 18th-century British mathematician Thomas Bayes, provides a way to update the probability of an event based on new evidence or information, describing conditional probability.
Basic Formulation
The basic form of Bayes' theorem is:
$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$
Where:
- $P(A|B)$ is the probability of A given B
- $P(B|A)$ is the probability of B given A
- $P(A)$ is the probability of A
- $P(B)$ is the probability of B
This formula allows us to calculate the probability of an event A occurring, given that we know event B has occurred, by using our prior knowledge of the probabilities of A and B, and the probability of B occurring given A.
Application to Multiple Events
Students are expected to apply Bayes' theorem to multiple events. This extension of the basic formula involves considering multiple conditional probabilities.
For $n$ events, Bayes' theorem is expressed as
$$P(B_i|A) = \frac{P(B_i)P(A|B_i)}{\displaystyle{\sum^{n}_{k=0} P(B_k) P(A | B_k)}}$$
where
- $B_i$ is the desired event
- $A$ is a given event
Although this looks daunting, we can break this down easily: The numerator describes the probability of both $A$ and $B_i$ has happened , and you know $A$ has already happened, and the denominator updates the sample space with the total probabilities that lead to $A$ happening.
Suppose we have a medical test for a rare disease. Let:
- A: Patient has the disease
- B: Test result is positive
Given:
- P(A) = 0.01 (1% of the population has the disease)
- P(B|A) = 0.95 (95% chance of a positive test if you have the disease)
- P(B|not A) = 0.10 (10% chance of a false positive)
We can calculate P(A|B), the probability of having the disease given a positive test:
$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B|A) \cdot P(A) + P(B|not A) \cdot P(not A)}$$
$$= \frac{0.95 \cdot 0.01}{0.95 \cdot 0.01 + 0.10 \cdot 0.99} \approx 0.0876$$
So, even with a positive test, there's only about an 8.76% chance of actually having the disease.
Students often confuse independence with mutual exclusivity. Independent events can occur together and their probabilities do not affect each other, while mutually exclusive events cannot.
Medical Risk Assessment
One of the most practical applications of Bayes' theorem is in medical risk assessment. It allows healthcare professionals to update the probability of a patient having a certain condition based on test results or symptoms.
Consider a screening test for a genetic disorder:
- A: Patient has the disorder
- B: Test is positive
- C: Patient has a family history of the disorder
Given:
- P(A) = 0.001 (0.1% prevalence in general population)
- P(B|A) = 0.99 (99% sensitivity)
- P(B|not A) = 0.02 (2% false positive rate)
- P(C|A) = 0.30 (30% of those with the disorder have a family history)
- P(C) = 0.05 (5% of the population has a family history)
We can use Bayes' theorem to calculate P(A|B,C), the probability of having the disorder given a positive test and family history:
$$P(A|B,C) = \frac{P(B,C|A) \cdot P(A)}{P(B,C)}$$
$$= \frac{P(B|A) \cdot P(C|A) \cdot P(A)}{P(B,C)}$$
(Assuming B and C are conditionally independent given A)
$$\approx 0.0294$$
This means there's about a 2.94% chance of having the disorder given these conditions.
When solving complex Bayes' theorem problems, it often helps to draw a probability tree or use a contingency table to organize the given information.
