Question

There are 3 clubs A,B,C in a town with 40,50,60 members respectively, 10 people members of all the three clubs, 70 members belong to one club. A member is randomly selected. Find the probability that he has membership of exactly two clubs.

Answer 1

Hint : In the solution, firstly find the number of members there in each club that is left and take the ratio of the remaining members in each of the total number of members in the three clubs A, B, and C to find the probability of the membership of exactly two clubs. The total members can be found by adding all 3 club members.

Complete step-by-step answer :
The number of clubs is $x = 3$
The total members in the three clubs are $y = 40 + 50 + 60 = 150$
The number of members that belong to one club is 70
The remaining members is,

z = 150 - 70\\\ = 80 \end{array}$$ Number of peoples that are members in all the 3 clubs is 10 Then the memberships in 3 clubs will be $$\begin{array} p = 3 \times 10\\\ p = 30 \end{array}$$ Now the remaining members are $$80 - 30 = 50$$ Now the each of the two clubs has 50 members, and each club has $$q = \dfrac{{50}}{2} = 25$$ So the equation to find the probability of the members of exactly two clubs is $$P = \dfrac{q}{x}$$ Substituting the values in the above equation, then we will get $$P = \dfrac{q}{x}$$ $$\begin{array} = \dfrac{{25}}{{150}}\\\ = \dfrac{1}{6} \end{array}$$ Therefore, the probability that he has a membership of exactly two clubs is $$\dfrac{1}{6}$$ **Note** : Here, in the solution, while finding the probability, we have to remember that the outcomes of the trials should be place in the numerator and the total number of trails that we have done will be placed in the denominator, then we can get the probability that he has the membership of exactly two clubs. It is given that 10 people are the members of all the three clubs, then we have to take the product of 10 with the 3 to know the total memberships filled all.