Question

Kansas City, despite its name, sits on the border of two US states: Missouri and Kansas. The Kansas City metropolitan area consists of 15 counties, 9 in Missouri and 6 in Kansas. The entire state of Kansas has 105 counties and Missouri has 114. Use Bayes’ theorem to calculate the probability that a relative who just moved to a county in the Kansas City metropolitan area also lives in a county in Kansas. Make sure to show P(Kansas) (assuming your relative lives either in Kansas or Missouri), P(Kansas City metropolitan area), and P(Kansas City metropolitan area | Kansas).

Answer 1

Hopefully it is pretty clear that there are 15 counties in the Kansas City metro area, and 6 of them are in Kansas, so the probability of being in Kansas, given you know someone lives in the Kansas City metro area, should be $6 / 15$, which is equivalent to $2 / 5$. The purpose of this question, however, is not just to get an answer but to show that Bayes' theorem provides the tools to solve it. When we work on harder problems, it will be very helpful to have established trust in Bayes' theorem. So, to solve $P$ (Kansas I Kansas City), we can use Bayes' theorem as follows:

$P($ Kansas $\mid$ Kansas City $)=\frac{P(\text { Kansas City } \mid \text { Kansas }) \times P(\text { Kansas })}{P(\text { Kansas City })}$

From our data we know that of the 105 counties in Kansas, 6 are in the Kansas City metro area:
$P($ Kansas City $\mid$ Kansas $)=\frac{6}{105}$
And between Missouri and Kansas there are 219 counties, 105 of which are in Kansas:
$$
P(\text { Kansas })=\frac{105}{219}
$$
And of this total of 219 counties, 15 are in the Kansas City metro area:
$$
P(\text { Kansas City })=\frac{15}{219}
$$

Filling in all of the parts of Bayes' theorem, then, gives us:
$$
P(\text { Kansas } \mid \text { Kansas City })=\frac{\frac{6}{105} \times \frac{105}{219}}{\frac{15}{219}}=\frac{2}{5}
$$