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 (a) you both enter the same sections? (b) you both enter the different sections?

Answer 1

My friend and I are among the $100$ students.
Total number of ways of selecting $2$ students out of $100$ students =100_{C2}$$=100_{C2}.
(a) The two of us will enter the same section if both of us are among $40$ students or among $60$ students.
∴ The number of ways in which both of us enter the same section =$40_{C2}+60_{C2}$
∴ The probability that both of us enter the same section
$=\dfrac{40_{C2}+60_{C2}}{100_{C2}}$

$=\dfrac{(39×40)+(59×60)}{99×100}=\dfrac{17}{33}$

(b) $P$(we enter different sections)
= $1 - P$(we enter the same section)
$=1-\dfrac{17}{33}=\dfrac{16}{33}$