College Algebra, Chapter 10, 10.3, Section 10.3, Problem 68

What is the probability that in a group of six students at least two have birthdays in the same month?

It is reasonable to assume that the six birthdays are independent and that each month has 30 days. Let $E$ be the event that two of the students have birthdays in the same month. Then we consider the complimentary event $E'$, that is, that no two students have birthdays in the same month. To find this probability, we consider the students one at a time. The probability that the first student has a birthday is 1, the probability that the second has a birthday different from the first is $\displaystyle \frac{29}{30}$, the probability that the third has a birthday different from the first two is $\displaystyle \frac{29}{30}$ and so on. Thus,

$\displaystyle P(E') = 1 \cdot \frac{29}{30} \cdot \frac{28}{30} \cdot \frac{27}{30} \cdot \frac{26}{30} \cdot \frac{25}{30} \cdot \frac{24}{30} \cdot \frac{23}{30} = 0.359686$

So,

$P(E) = 1- P(E') = 1-0.359686 = 0.640314$