The probability that out of 10 persons, all born in June, at least two have the same birth day is

$\frac{30^{10}-{^{30}C}_{10}}{(30)^{10}}$

Since each person can have any one of the thirty days of June month as his (her) birthday. Therefore, Number of ways in which 10 persons can have birthdays in the month of June

$= 30 ×30 × 30 × .......× 30 (10 \, times) = 30^{10}$

∴ Required probability

= 1- Probability that no two persons have the same birth day

$=1 - \frac{^{30}C_{10}}{30^{10}}= \frac{30^{10}-{^{30}C}_{10}}{(30)^{10}}$