Obviously if there are 8 or more people or more 
in the room the probability is 1.

We go for the complement event, the event that n people 
all have different birth days.  Then subtract from 1.
[Notice that "birth day" does not mean "birthday". LOL]

Case 1: One person in the room.  

The probability is 0 that 2 have the same birth day, for 
there aren't 2 people in the room.

Case 2: Two people in the room.

We can assign them all different birth days in 7P2 = 42 ways.
We can assign them birthdays in 7*7=49 ways.  The 
probability that they have different birth days is 
42/49. So the probability that they have the same
birthday 1-42/49 = 1/7 = 0.142871429

Case 3: Three people in the room.

We can assign them all different birth days in 7P3 = 210 ways.
We can assign them birthdays in 7*7*7=343 ways.  The 
probability that they have different birth days is 
210/343. So the probability that they have the same
birthday 1-210/343 = 19/49 = 0.387755102
  
Case 4: Four people in the room.

We can assign them all different birth days in 7P4 = 840 ways.
We can assign them birthdays in 7*7*7*7=2401 ways.  The 
probability that they have different birth days is 
840/2401. So the probability that they have the same
birthday 1-840/2401 = 223/343 = 0.6501457726

That's the answer, 4 people, but let's follow through
for instructive purposes. 

Case 5: Five people in the room.

We can assign them all different birth days in 7P5 = 2520 ways.
We can assign them birthdays in 7*7*7*7*7=16807 ways.  The 
probability that they have different birth days is 
840/2401. So the probability that they have the same
birthday 1-2520/16807 = 2041/2401 = 0.850062474

Case 6: Six people in the room.

We can assign them all different birth days in 7P6 = 5040 ways.
We can assign them birthdays in 7*7*7*7*7*7=117649 ways.  The 
probability that they have different birth days is 
5040/117649. So the probability that they have the same
birthday 1-5040/117649 = 112609/117649 = 0.9571607060

Case 7: Seven people in the room.

We can assign them all different birth days in 7P7 = 5040 ways.
We can assign them birthdays in 7*7*7*7*7*7*7=823543 ways.  The 
probability that they have different birth days is 
5040/2401. So the probability that they have the same
birthday 1-5040/823543 = 116929/117649 = 0.993880101

Edwin