Bayes' Theorem

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In this article, I will explain Bayes’ Theorem, which is the core of the Bayesian statistics, with a simple example.

First, the theorem is given as:

$P(A|B) = \frac{P(B | A)\, P(A)}{P(B)} \text{ where } P(B) = {\sum_i P(B|A_i) P(A_i)} \neq 0$

And here is the situation:

Suppose two companies, A and B, manufacture 4.6L SOHC V-6 engines that go into a Mustang. Ford bought 5000 engines from those two companies, and

9 out of 1000 engines from company A were defective.

25 out of 4000 engines from company B were defective.

Now, the question is:

What is the probability that a defective engine came from company B?

Define some variables we need:

A = the engine came from company A
B = the engine came from company B
D = the engine is defective

Let’s write down the original question in a bit more formal way. What is the probability of B given D? In other words, what is the probability that the engine came from company B given it is defective?

Here is the conditional probability of getting a defective engine from company A:

$P(D|A) = \frac{9}{1000} = 0.009$

Likewise, the conditional probability of getting a defective engine from company B is:

$P(D|B) = \frac{25}{4000} = 0.0625$

The prior probability of an engine coming from company A is:

$P(A) = \frac{1000}{5000} = 0.2$

Similarly for B:

$P(B) = \frac{4000}{5000} = 0.8$

Now we have what we need to calculate $P(B

D)$.

$P(B|D) = \frac{P(B)P(D|B)}{P(A)P(D|A) + P(B)P(D|B)} = \frac{(0.8)(0.0625)}{(0.2)(0.009) + (0.8)(0.0625)} \approx 0.965$

So, when Ford finds a defective engine, it’s most likely coming from company B.