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 $={}^{100}{C}_2$

(a) The two of us will enter the same section if both of us are among $40$ students or among $60$ students

$\therefore$ Number of ways in which both of us enter the same section = ${}^{40}{C}_2{+}^{60}{C}_2$

$\therefore$ Probability that both of us enter the same section

$=\frac{{}^{40}{C}_2{+}^{60}{C}_2}{{}^{100}{C}_2}=\frac{(39\times 40)+(59\times 60)}{99\times 100}=\frac{17}{33}$

(b) $P$ (we enter different sections)

$=1-$ $P$ (we enter the same section)

$=1-\frac{17}{33}=\frac{16}{33}$