Question

Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among the 100 students, what is the probability that

1. you both enter the same sections?
2. you both enter the different sections?

Answer 1

Let there be two sections A and B having 40 and 60 students, respectively.

a. i. Suppose both the students belong to section A.

∴ 38 students are selected out of 98 students.

Ways to select 38 students from 98 = ⁹⁸C₃₈

Ways to select 40 students from 100, without any condition n(S) = ¹⁰⁰C₄₀

Probability that both students (he and his friend) get admission in the same section A

= `(""^98C_38)/(""^100C_40)`

= `(98!)/(38!60!) xx (40!60!)/(100!)`

= `(98! xx 40! xx 60!)/(38!60! xx 100.99(98!))`

= `(40.39)/(100 xx 99)`

= `26/165`

ii. If both students get admission in section B. Then number of ways to select 58 students from 98 = ⁹⁸C₅₈

number of ways to select 60 students from 100 = ¹⁰⁰C₆₀

So, if those students get admission in section B then the probability is

= `""^98C_58 ÷ ""^100C_60`

= `(98!)/(58!40!) ÷ (100!)/(60!40!)`

= `(98!)/(58!40!) xx (60 xx 59 xx (58!) xx (40!))/(100 xx 99 xx 98!)`

= `(60.59)/(100.99)`

= `59/(5 xx 33)`

= `59/165`

Both the students get admission in section A or B Then his probability

= `26/165 + 59/165`

= `85/165`

= `17/33`

b. Probability of both students getting admission in different sections

= `1 - 17/33`

= `(33 - 17)/33`

= `16/33`