# Bayes' Theorem

*Bayes' theorem* (or *Bayes' Law* and sometimes *Bayes' Rule*) is a direct application of conditional probabilities. The probability P(A|B) of "A assuming B" is given by the formula

P(A|B) = P(A∩B) / P(B)

which for our purpose is better written as

P(A∩B) = P(A|B)·P(B).

The left hand side P(A∩B) depends on A and B in a symmetric manner and would be the same if we started with P(B|A) instead:

P(B|A)·P(A) = P(A∩B) = P(A|B)·P(B).

This is actually what Bayes' theorem is about:

(1) | P(B|A) = P(A|B)·P(B) / P(A). |

Most often, however, the theorem appears in a somewhat different form

(1') | P(B|A) = P(A|B)·P(B) / (P(A|B)P(B) + P(A|B)P(B)), |

where B is an event complementary to B:

This is because

A | = A ∩ (B ∪ B) |

= A∩B ∪ A∩B |

and, since A∩B and A∩B are mutually exclusive,

P(A) | = P(A∩B ∪ A∩B) |

= P(A∩B) + P(A∩B) | |

= P(A|B)P(B) + P(A|B)P(B). |

More generally, for a finite number of mutually exclusive and exhaustive events H_{i}

H_{k} ∩ H_{m} = Φ, for k ≠ m and

H_{1} ∪ H_{2} ∪ ... ∪ H_{n} = Ω,

Bayes' theorem states that

P(H_{k}|A) = P(A|H_{k}) P(H_{k}) / ∑_{i} P(A|H_{i}) P(H_{i}),

where the sum is taken over all i = 1, ..., n.

We shall consider several examples.

**Example 1. Monty Hall Problem.** [Havil, pp. 61-63]

Let A, B, C denote the events "the car is behind door A (or #1)", "the car is behind the door B (or #2)", "the car is behine the door C (or #3)." Let also M_{A} denote the event of Monty opening door A, etc.

You are called on stage and point to door A, say. Then

_{B}|A) = 1/2 | because Monty has to choose between two carless doors, B and C |

P(M_{B}|B) = 0, | because Monty never opens the door with a car behind |

P(M_{B}|C) = 1, | for the very same reason that _{B}|B) = 0 |

Since A, B, C are mutually exclusive and exhaustive,

P(M_{B}) | = P(M_{B}|A)P(A) + P(M_{B}|B)P(B) + P(M_{B}|C)P(C) |

= 1/2 × 1/3 + 0 × 1/3 + 1 × 1/3 | |

= 1/2. |

Now you are given a chance to switch to another door, B or C (depending on which one remains closed.) If you stick with your original selection (A),

P(A|M_{B}) | = P(M_{B}|A)P(A)/P(M_{B}) |

= 1/2 × 1/3 / 1/2 | |

= 1/3. |

However, if you switch,

P(C|M_{B}) | = P(M_{B}|C)P(C)/P(M_{B}) |

= 1 × 1/3 / 1/2 | |

= 2/3. |

You'd be remiss not to switch.

**Example 2. Sick Child and Doctor**. [Falk, p. 48]

A doctor is called to see a sick child. The doctor has prior information that 90% of sick children in that neighborhood have the flu, while the other 10% are sick with measles. Let F stand for an event of a child being sick with flu and M stand for an event of a child being sick with measles. Assume for simplicity that

A well-known symptom of measles is a rash (the event of having which we denote R).

Upon examining the child, the doctor finds a rash. What is the probability that the child has measles?

**Example 3. Incidence of Breast Cancer**. [von Savant, pp. 103-104]

In a study, physicians were asked what the odds of breast cancer would be in a woman who was initially thought to have a 1% risk of cancer but who ended up with a positive mammogram result (a mammogram accurately classifies about 80% of cancerous tumors and 90% of benign tumors.) 95 out of a hundred physicians estimated the probability of cancer to be about 75%. Do you agree?

### References

- R. B. Ash,
*Basic Probability Theory*, Dover, 2008 - R. Falk,
*Understanding Probability and Statistics*, A K Peters, 1993 - J. Havil,
*Impossible?*, Princeton University Press, 2008 - M. vos Savant,
*The Power of Logical Thinking*, St. Martin's Press, NY 1996

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#### Sick Child and Doctor (Solution)

P(M|R) | = P(R|M) P(M) / (P(R|M) P(M) + P(R|F) P(F)) | |

= .95 × .10 / (.95 × .10 + .08 × .90) | ||

≈ 0.57. |

Which is nowhere close to 95% of P(R|M).

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#### Incidence of Breast Cancer (Solution)

Introduce the events:

P | - mammogram result is positive, | |

B | - tumor is benign, | |

M | - tumor is malignant. |

Bayes' formula in this case is

P(M|P) | = P(P|M) P(M) / (P(P|M) P(M) + P(P|B) P(B)) | |

= .80 × .01 / (.80 × .01 + .10 × .99) | ||

≈ 0.075 | ||

= 7.5%. |

A far cry from a common estimate of 75%.

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