Q. 2.13 - A First Course in Probability

Question

A certain town with a population $100,000$has $3$newspapers:$I, II,$ and $III.$ The proportions of townspeople who read these papers are as follows:

$I: 10$ percent $I$ and $II: 8$ percent$I$,$II$ and $III: 1$ percent

$II: 30$ percent$I$ and $III: 2$percent

$III: 5$percent$II$ and$III: 4$ percent

(The list tells us, for instance, that$8000$ people read newspapers $I$and$II$)

$\left(a\right)$ Find the number of people who read only one newspaper.

$\left(b\right)$ How many people read at least two newspapers?

$\left(c\right)$ If$I$ and $III$are morning papers and $II$is an evening paper, how many people read at least one-morning paper plus an evening paper?

$\left(d\right)$How many people do not read any newspapers?

$\left(e\right)$How many people read the only one-morning paper and one evening paper?

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right)20,000 \\ \left(b\right)12,000 \\ \left(c\right)11,000 \\ \left(d\right)68,000 \\ \left(e\right)10,000 \end{gathered}$$

Step 1 Given Information.

A certain town with a population $100,000$has $3$newspapers:$I, II,$ and$III$. The proportions of townspeople who read these papers are as follows:

$I: 10$percent $I$and$II: 8$ percent$I$,$II$ and $III: 1$ percent

$II: 30$ percent$I$ and $III: 2$ percent

$III: 5$ percent$II$ and$III: 4$ percent

Step 2 Part (a) Explanation.

$I$only $+II$only $+III$only$1000+19000+0=20000$.

 

Step 3 Part (b) Explanation.

$( I$ and$II$ only $)+(II$and $III$only $)+(I$and $III$only$)+(I$, $II$ and$III)$$=7,000+3,000+1,000+1,000=12,000$.

Step 4 Part (c) Explanation.

$(I$and$II$ only$) + ( III$ and$II$ only)$+(\mathrm{I},\mathrm{II}$and $II$only$)=7,000+3,000+1,000=11,000$.

Step 5 Part (d) Explanat\ion.

From Part (a) we have$68000$.

Step 6 Part (e) Explanation.

 $(I$ and$II$ only $) + (III$and$II$ only $)$ $=7,000+3,000=10,000$.