Question

There are three clubs $A,B,C$ in a town with $40,50,60$ members respectively $10$ people are members of all the three clubs, $70$ members belong to only one club.

A member is randomly selected.

Find the probability that he has a membership of $exactlytwoclubs.$

Answer 1

Step-1: Write down the given information

From the Venn diagram

$A=a+e+g+d=40$…………(i)

$B=b+e+g+f=50$ ………...(ii)

$C=c+d+g+f=60$ …………..(iii)

Number of members of all the three clubs, $g=10$

Number of members belong to only one club, $a+b+c=70$

Step-2: Find the number of people belonging to exactly two groups

Add$\left(i\right),\left(ii\right)and\left(iii\right)$

$a+e+g+d+b+e+g+f+c+d+g+f=40+50+60$

$a+b+c+2e+2d+2f+3g=40+50+60$

$70+2(d+e+f)+30=150$

$70+30+2(d+e+f)=150$

$100+2(d+e+f)=150$

$2(d+e+f)=150-100$

$(d+e+f)=\frac{50}{2}$

Total number of people belonging to exactly two groups; $(d+e+f)=25$

Step-3: Find the total number of people in the club

total number of people in the club=$a+b+c+d+e+f+g=70+25+10$

total number of people in the club= $a+b+c+d+e+f+g=105$

Step-4: Find the probability that he has a membership of $exactlytwoclubs.$

The probability that he has a membership of $exactlytwoclubs.$$\frac{Totalnumberofpeoplebelongingtoexactlytwogroups}{totalnumberofpeopleintheclub}=\frac{25}{105}$$=\frac{5}{21}$

Hence, the probability that he has a membership of $exactlytwoclubs$is $\frac{5}{21}$