# Stuck on this birthday problem

Assume for simplicity that N people, all born in April (a month of 30 days), are collected in a room Consider the event of at least two people in the room being born on the same date of the month, even if in different years, e.g. 1980 and 1985. What is the smallest N so that the probability of this event exceeds 0.5?

• What have you tried? I would try with calculating what is the probability that exactly 2 people share the same date of month, then 3 and then 4 ... – Matti P. Oct 29 '18 at 7:55
• I tried to do it this way, prob. of at least two people sharing the same birthday = 1 - no couple of persons sharing the same birthday. – user17616 Oct 29 '18 at 7:57
• But I'm stuck in the calculation part as it involves factorials. – user17616 Oct 29 '18 at 7:58
• @user17616 You are on the right track. Do this for $N=2,3,4$, and you should see a pattern. – Per Manne Oct 29 '18 at 8:01
• 1 - 30! / { 30^n*(30-n)! } = 0.5. How to figure out n from this? – user17616 Oct 29 '18 at 8:01

From the article linked above, we use the same strategy, and we get that $$\frac{30}{30} \times \frac{29}{30} \times \frac{28}{30} \times \frac{27}{30} \times \frac{26}{30} \times \frac{25}{30} \times \frac{24}{30}\le0.5$$ The answer follows.