There are 20 people in a room what is the probability that they all have different birthdays

What we're saying here is that for every different date in the year, one person must have that birthday, and that includes one person who has a birthday on February 29th!

It's February 29th that gives the problems. Most years have 365 days in them and February only has 28 days. But every fourth year is a leap year and there is an extra day in February making a total of 366 days. Therefore we need to base our sums on the total number of days in four years which is: 3 x 365 + 366 = 1461.

• Let's suppose that the first person to come into our room happens to be the person who's birthday is Feb 29th. The chances of that are 1/1461.
• When the second person comes in, the only date they can't have a birthday on is Feb 29th, so the chance of the second person being different is 1460/1461.
• Now let's say, for example, the second person's birthday was on October 5th. Out of a four-year period of 1461 days there are four October 5ths. Therefore when the third person comes into the room there are a total of FIVE days they can't have a birthday on. (That's four Oct 5ths and one Feb 29th.) So the third person's chance of being different is 1456/1461. The chances of all three being different are:
  1/1461 x 1460/1461 x 1456/1461
• When the fourth person comes in, their chance of a different birthday is 1452/1461, and for the fifth person it's 1448/1461.
• By the time the 366th person has come into the room, if everybody else has had a different birthday then there will only be one calendar date left e.g. July 24th. In a period of 1461 days, there are only four of these, so the 366th person's chances of being different from everybody else are 4/1461.
• Multiply the whole lot together! You get:
  1/1461 x 1460/1461 x 1456/1461 x ... all the way to ... x 8/1461 x 4/1461
• There are 366 fractions here all with 1461 on the bottom, so this all simplifies to:
  (1 x 1460 x 1456 x 1452 x ... 8 x 4)/1461366. But of course we don't need to put in the 1x at the front, so it becomes:
  (1460 x 1456 x 1452 x ... 8 x 4)/1461366
• There are 365 numbers in the bracket and they will each break down like this: 1460 = 365 x 4 and 1456 = 364 x 4 and 1452 = 363 x 4 and so on. So if we divide every term in the bracket by 4 and bring all the 4's outside we get:
  4365 x (365 x 364 x 363 x ... all the way to ... x 3 x 2 x 1)/1461366
  The bit in the bracket is a factorial which we can write like this: 365!
• So putting the whole lot together the chances of the first person in the room having a birthday on Feb 29th, and then the other 365 people all being different are:
  (4365 x 365! ) / 1461366
• Finally... we have to allow for the fact that our Feb 29th birthday person could come into the room at any time, rather than having to be first. Supposing everyone was lined up to get into the room. How many different positions in the line could he stand in? The answer is 366. So we multiply our last answer by 366 and at last we get the answer we're looking for:
  (366 x 4365 x 365! ) / 1461366
• We can make this a tiny bit neater, because 366 x 365! = 366! So at last, here is the sum to work out the chances of 366 strangers in a room all with different birthdays:
  
  
• This works out to be about 1 chance in 3520000000000000000000000000000000000000000000000000000000000000000000000000000
  0000000000000000000000000000000000000000000000000000000000000000000000000000000.
  (There are 155 zeros there.)