1. probability

There are thirty people in a room and all of them write down the day and month of their birth.

a)What is the probability that two or more of these dates will be the same (remember, just day and month, not year)?

b)What is the probability that three or more of these dates will all coincide?

2. Hello, perash!

This is a classic . . . The Birthday Paradox.
. . I'll do the easy one.

Thirty people write down the day and month of their birth.

a) What is the probability that 2 or more of these dates will be the same ?

The opposite of "at least one match" is "no matches."

Person #1 can have any birthday: . $\frac{365}{365}$

#2 can have any of the remaining 364 days: . $\frac{364}{365}$

#3 can have any of the remaining 363 days: . $\frac{363}{365}$

. . . and so on . . .

#30 can have any of the remaining 336 days: . $\frac{336}{365}$

$P(\text{no match}) \:=\:\frac{365}{365}\cdot\frac{364}{365}\cdot\frac {363}{365}\cdots \frac{336}{365}$

Hence: . $P_0 \;=\;\frac{365!}{335!365^{30}} \;=\;\frac{{365\choose335}}{365^{30}}$

Therefore: . $P(\text{2 or more}) \;=\;1 - P_0$

. . (I'll let you crank it out.)
.