Question

The following data were given in a study of a group of$1000$subscribers to a certain magazine: In reference to the job, marital status, and education, there were $312$professionals, $470$married persons, $525$college graduates, $42$professional college graduates, $147$married college graduates, $86$married professionals, and $25$married professional college graduates. Show that the numbers reported in the

the study must be incorrect.

Hint: Let $M,W,$and$G$denote, respectively, the set of professionals, married persons, and college graduates. Assume that one of the $1000$persons is chosen at random, and use Proposition $4.4$to show that if the given numbers are correct, then$P(M\cup W\cup G)>1$.

Short answer

We have$P(M\cup W\cup G)>1$.

Step-by-step solution

Given Information.

Marital status, and education, there were $312$professionals, $470$married persons, $525$college graduates, $42$professional college graduates, $147$married college graduates,$86$married professionals, and$25$married professional college graduates.

Explanation.

Using the proposition $4.4$given in the book, we expand the equation and find that the probability calculated on the left side turns out to be greater than$1$, which is not allowed, and hence, the numbers reported must be wrong.

Explanation.

$$\begin{gathered} P(M\cup W\cup G)=P\left(M\right)+P\left(W\right)+P\left(G\right)-P(MW)-P(WG)-P(MG)+P(MWG) \\ =0.312+0.470+0.525-0.086-0.147-0.042+0.025 \\ =1.057 \\ >1 \end{gathered}$$