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?

Solution :

My friend and I are amongst the 100 students.

Total number of ways of selecting 2 students out of 100 students = ${}^{100}\mathrm{C}_2 = \frac{100\times 99\times 98!}{98! \times 2!} = 50 \times 99 = 4950$

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

∴ Number of ways in which both of us enter the same section = ${}^{40}\mathrm{C}_2 + {}^{60}\mathrm{C}_2 = \frac{40\times 39\times 38!}{38! \times 2!} + \frac{60\times 59\times 58!}{58! \times 2!} = 2550$

∴ Probability that both of us enter the same section = $\frac{\mathrm{No}. \mathrm{of} \mathrm{ways} \mathrm{in} \mathrm{which} \mathrm{both} \mathrm{of} \mathrm{us} \mathrm{enter} \mathrm{same} \mathrm{section}}{\mathrm{Total} \mathrm{ways} \mathrm{of} \mathrm{selecting} 2 \mathrm{students} \mathrm{out} \mathrm{of} 100}$

= $\frac{2550}{4950} = \frac{17}{33}$

(b) P(we enter different sections)

= 1 − P(we enter the same section)

= $1 - \frac{17}{33} = \frac{16}{33}$