Person 1 can have any birthday, like today, Nov 20. So the probability of 1 person being born on any given day is obviously 100%. We will write 100% as 365/365.
The probability that Person 2 has the same birthday as Person 1 (i.e. today Nov 20) is 1/365.
To find the probability that both people were born on the same day means that you have to multiply the individual probabilites:
Now what about 3 people and this is where it starts to get counter intuitive because for person 3 there are two opportunities to be born on the same day, not 1.
With Person 3 they could share a birthday with either Person 2 or Person 1, so now 2 out of 365 days are in play.
So as you add more people, the number of days in play as their birthday day, increases and the probabilty starts to increase at a faster rate than you think.
To solve the birthday problem, we need to use one of the basic rules of probability:
the sum of the probability that an event will happen and the probability that the
event won't happen is always 1.
Pevent happens + Pevent doesn't happen = 1
P(two people share birthday) + P(no two people share birthday) =1
P(two people share birthday) = 1 -P(no two people share birthday)
So, what is the probability that no two people share a birthday?
Person 1 has a birthday on some given day. Again, choose today: Nov 20
Now we have specified that Person 2's birthday cannot be today. So that means person 2 was born on some day among the 364 remaining days. So the chance that two people have different birthdays is:
Now add Person 3. There are 363 days available for person 3 to be born so as not to be born on the same day as Person 1 or Person 2.
The probability that these 3 people have three different birthdays must then be:
So now we see the pattern. Adding a 4th person would mean we need a 362/365 term in the above. That would yield 98.36% that no two persons were born on the same day.
The probability that, if there are 4 people in the room, that two of them were born on the same day is then 1 -98.36 = 1.64%
Your exercise in Excel is to now use this probability logic to determine the value of N persons in which the probability of 2 people being born on the same day is 50% (or inversely the probability that all N people were born on different days is 50%).