Essentials of Mathematical Probability and Statistics

Section 4.6 Bayes' Theorem

Conditional probabilities 4.5.3 can be computed using the methods developed above if the appropriate information is available. Some times you will however have some information available, such as \(P(A | B)\) but need \(P(B | A)\text{.}\) The ability to "play around with history" by switching what has been presumed to occur leads to an important result known as Bayes' Theorem.

To illustrate this result, from the web site http://stattrek.com/probability/bayes-theorem.aspx consider the following problem:

Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent years, it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. When it doesn't rain, he incorrectly forecasts rain 10% of the time. What is the probability that it will rain on the day of Marie's wedding?

Notice, all days can be classified into one of two disjoint options:

• Rainy, in which case we can deduce from the given info that P(Rain) = 5/365
• Not Rainy, and since this is the complement of above, P(Not Rain) = 360/365

In the notation of Bayes Theorem 4.6.1, let A represent a forecast of Rain and note you have

and

Further, you are given the conditional probabilities

Notice that the question provided requests that you find the probability of Rain given that the weatherman has forecasted rain. What is given on the other hand is the reverse of that conditional probability. Using Bayes' Theorem allows you to turn this around...

Hence, putting these together gives

So, normally there is only approximately a 1.369 percent chance of rain (5/365) on a given day but given that the weatherman has forecast rain, the chance of rain increases to a little more than 11 percent.

Here's a standard and yet surprising example:

Your neighbor has 2 children. You learn that he has a daughter Anna. What is the probability that Anna’s sibling is a sister? Your first reaction might be that it is obviously 1/2 since (ok, roughly) it is equally likely that the other sibling was born female or male. But is that accurate?

If you were to list all of the possible female/male outcomes when having two children, then the sample space is S = {FF, FM, MF, MM} with FM meaning the first child is female and the second is male. Assuming again that girls and boys are equally likely to be born, these 4 outcomes have equal probability of \(1/4\text{.}\)

The question asks is whether the neighbor has another daughter in the set Fem2 = {FF} but since Anna is a girl then the possible outcomes now is only Fem = {FF, FM, MF}.

So,

Bayes' Theorem 4.6.1 also works well in analyzing "Let's Make A Deal" finale choice.

The interactive cell below can be used to easily compute all of the conditional probabilities associated with Bayes' Theorem 4.6.1. Notice how the relative size of the pie-shaped partition changes when you presume that an event in the space has already occurred.

You can actually also use Bayes' Theorem 4.6.1 to answer a easy question! Indeed, suppose that you draw one card from a shuffled and standard 52 card deck. Given that you know the card is an Ace, what is the probability that it is also a Heart.

Using the Bayes' approach, let's break up the world into Hearts (H) and non-Hearts (N). Easily,

and so by Bayes'

as expected!