Q. 2.14 - A First Course in Probability

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

Step-by-Step Solution

Answer

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

Step 1 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.

Step 2 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.

Step 3 Explanation.

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