Example 31

EXAMPLE 31 Bayes' Rule

The Gardasil vaccine

Table 24 contains a contingency table (crosstabulation) of the variables com-pleted and insurance type for the data set Gardasil.¹²

Table 5.57: Table 24 Contingency table of the variables completed and insurance type

			Insurance type
		Hospital- based	Medical assistance	Military	Private payer	Total
Completed	No	45	220	209	470	944
	Yes	39	55	122	253	469
	Total	84	275	331	723	1413

Define the following events:

$M : Insurance type = Military$
$C : Completed the vaccination treatment$

Use Bayes’ Rule to find the probability that a randomly selected patient used military insurance, given that he or she completed the treatment.

Solution

We are interested in the probability $P (M given C)$ . Substituting $M$ for $A$ and $C$ for $B$ in the formula for Bayes’ Rule, we obtain:

$P (M given C) = \frac{P (M) \cdot P (C given M)}{P (M) \cdot P (C given M) + P (M^{C}) \cdot P (C given M^{C})}$

From Table 24, we have $P (M) = \frac{331}{1413} and P (C given M) = \frac{122}{331}$ . The event $M^{C}$ consists of all other insurance types besides military. So we have $P (M^{C}) = \frac{1082}{1413}$ , and $P (C given M^{C}) = \frac{347}{1082}$ . Thus,

$P (M given C) = \frac{331}{1413} \cdot \frac{122}{331} \frac{331}{1413} \cdot \frac{122}{331} + \frac{1082}{1413} \cdot \frac{347}{1082} = \frac{122}{1413} \frac{122}{1413} + \frac{347}{1413} = \frac{122}{469} \approx 0.26$

Note that this result is confirmed by the direct calculation of $P (M / C)$ using Table 24, which is the single cell of 122 military completers divided by the total number of completers.

NOW YOU CAN DO

Exercises 105–108.