Page 491 - Discrete Mathematics and Its Applications

470 7 / Discrete Probability

Before we had any extra information, we assumed that the probability that Bob selected the first
box was 1/2. However, with the extra information that the ball selected at random is red, this
probability has increased to approximately 0.645. That is, the probability that Bob selected a
ball from the first box increased from 1/2, when no extra information was available, to 0.645
once we knew that the ball selected was red. ▲

Using the same type of reasoning as in Example 1, we can find the conditional probability
that an event F occurs, given that an event E has occurred, when we know p(E | F), p(E | F),
and p(F). The result we can obtain is called Bayes’ theorem; it is named after Thomas Bayes,
an eighteenth-century British mathematician and minister who introduced this result.

THEOREM 1 BAYES’ THEOREM Suppose that E and F are events from a sample space S such that
p(E) = 0 and p(F) = 0. Then

p(E | F)p(F)
p(F | E) = .
p(E | F)p(F) + p(E | F)p(F)

Proof: The definition of conditional probability tells us that p(F | E) = p(E ∩ F)/p(E)
and p(E | F) = p(E ∩ F)/p(F). Therefore, p(E ∩ F) = p(F | E)p(E) and p(E ∩ F) =
p(E | F)p(F). Equating these two expressions for p(E ∩ F) shows that

p(F | E)p(E) = p(E | F)p(F).

Dividing both sides by p(E), we find that

p(E | F)p(F)
p(F | E) = .
p(E)

Next, we show that p(E) = p(E | F)p(F) + p(E | F)p(F). To see this, first note
that E = E ∩ S = E ∩ (F ∪ F) = (E ∩ F) ∪ (E ∩ F). Furthermore, E ∩ F and E ∩ F
are disjoint, because if x ∈ E ∩ F and x ∈ E ∩ F, then x ∈ F ∩ F =∅. Consequently,
p(E) = p(E ∩ F) + p(E ∩ F). We have already shown that p(E ∩ F) = p(E | F)p(F).
Moreover, we have p(E | F) = p(E ∩ F)/p(F), which shows that p(E ∩ F) =
p(E | F)p(F). It now follows that

p(E) = p(E ∩ F) + p(E ∩ F) = p(E | F)p(F) + p(E | F)p(F).

To complete the proof we insert this expression for p(E) into the equation p(F | E) =
p(E | F)p(F)/p(E). We have proved that

p(E | F)p(F)
p(F | E) = .
p(E | F)p(F) + p(E | F)p(F)

APPLYING BAYES’ THEOREM Bayes’ theorem can be used to solve problems that arise
in many disciplines. Next, we will discuss an application of Bayes’ theorem to medicine. In
particular, we will illustrate how Bayes’ theorem can be used to assess the probability that
someone testing positive for a disease actually has this disease. The results obtained from
Bayes’ theorem are often somewhat surprising, as Example 2 shows.