Bayes' Theorem/General Result

Theorem

Let $\Pr$ be a probability measure on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\set {B_1, B_2, \ldots}$ be a partition of the event space $\Sigma$.

Then, for any $B_i$ in the partition:

$\condprob {B_i} A = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\map \Pr A} = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\sum_j \condprob A {B_j} \map \Pr {B_j} }$

where $\ds \sum_j$ denotes the sum over $j$.

Proof

Follows directly from the Total Probability Theorem:

$\ds \map \Pr A = \sum_i \condprob A {B_i} \map \Pr {B_i}$

and Bayes' Theorem:

$\condprob {B_i} A = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\map \Pr A}$

$\blacksquare$

Examples

Arbitrary Example $1$

Let box $1$ contain $10$ good screws and $2$ unslotted screws.

Let box $2$ contain $8$ good screws and $4$ unslotted screws.

Let a box be selected at random.

Let a screw chosen from that box prove to be unslotted.

What is the probability that it came from box $2$?

Arbitrary Example $2$

Suppose that, in a population, $6$ out of every $1000$ people has an illness $X$.

It is known that:

if a person has $X$, there is a $92 \%$ probability that a blood test will be positive for $X$

if a person does not have $X$, there is a $0 \cdotp 5 \%$ probability that a blood test will be positive for $X$.

Let a person selected at random test positive for $X$.

What is the probability that this person actually has $X$?

Arbitrary Example $3$

Let $C_1$ be a a double-headed coin.

Let $C_2$ be a fair coin.

Let $A_1$ denote the event of tossing $C_1$.

Let $A_2$ denote the event of tossing $C_2$.

Suppose one of $C_1$ and $C_2$ is chosen at random with $\map \Pr {A_1} = \dfrac 1 2$ and $\map \Pr {A_2} = \dfrac 1 2$.

Let $B$ be the event of landing heads.

What is the probability that it was $C_1$ that was tossed?

Source of Name

This entry was named for Thomas Bayes.

Historical Note

Bayes' Theorem was published posthumously in $1763$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Bayes' theorem
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Bayes' theorem
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Bayes' Theorem
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Bayes' Theorem