>Rules of Probability>What Is Bayes' Theorem?

What Is Bayes' Theorem?

Bayes’ theorem tells you how to find $P (A ∣ B)$ when you already know $P (B ∣ A)$ , $P (A)$ and $P (B)$ . This is really useful in the real world, as the example below will show you. This is what Bayes’ theorem says:

Formula

Bayes’ theorem

P (A ∣ B) = \frac{P (A) \cdot P (B ∣ A)}{P (B)}

In many assignments, $P (B)$ will not be given. In that case you can find it in this way:

\begin{array}{l} P (B) & = P (A \cap B) + P (\overset{}{A} \cap B) \\ = P (A) \cdot P (B ∣ A) + P (\overset{}{A}) \cdot P (B ∣ \overset{}{A}) \end{array}

This is called the law of total probability.

Example 1

In the United States, lie detectors, or polygraphs, are used by the police during interrogations to try to determine whether a witness is telling the truth or not. In a study investigating lie detectors, it was found that:

If a person is lying, the probability of the lie detector uncovering the lie is $90 %$ .

If a person is telling the truth, there’s a $15 %$ chance that the lie detector thinks they’re lying.

A witness in a criminal case is connected to a lie detector, which claims that the witness is lying. If the probability of the witness lying is $25 %$ , what is the probability that the claim of the lie detector is correct?

This is an example where you’ll have to use Bayes’ theorem. Simply keep your head on straight and take it one step at a time.

1.

First, write down what you can tell from the information you’ve been given:

\begin{array}{l} L & = the person is lying, \\ T & = the person is telling the truth, \\ A_{L} & = the lie detector indicates a lie, \\ A_{T} & = the lie detector indicates truth . \end{array}

2.

You know that

P (L) = 0.25

. That makes

P (T) = 1 - P (L) = 0.75

3.

Furthermore, you know that

P (A_{L} ∣ L) = 0.90

and

P (A_{L} ∣ T) = 0.15

You want to find the conditional probability of $P (L ∣ A_{L})$ . This is where you use Bayes’ theorem, but first you’ll have to calculate $P (A_{L})$ using the law of total probability. You can insert the formula for this probability straight into Bayes’ theorem, but it might be easier to calculate it on its own first. You can do so like this:

\begin{array}{l} P (A_{L}) & = P (L) \cdot P (A_{L} ∣ L) + P (T) \cdot P (A_{L} ∣ T) \\ = 0.25 \cdot 0.90 + 0.75 \cdot 0.15 \\ = 0.3375 \end{array}

Then you can insert all the known numbers into Bayes’ theorem to find the answer.

\begin{array}{l} P (L ∣ A_{L}) & = \frac{P (L) \cdot P (A_{L} ∣ L)}{P (A_{L})} \\ = \frac{0.25 \cdot 0.90}{0.3375} \\ \approx 0.67 \end{array}

The probability of the witness actually lying when the lie detector says they’re lying is just $0.67$ , or 67%. That’s not particularly convincing!

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