Notes for March 13

Besides review the new topic, for today was another problem in probability. If we have a group of n people and none of them were born on Leap Day (February 29, very rare), what is the probability that at least two people have the same birthday?

Trying to do this problem directly is very difficult, so instead we try to figure out the probability of the opposite problem.

If we have n people and none of them are born on February 29, what is the probability of at least two people sharing a birthday? (Note: we are not asking for the same year, just the same day.)

Figuring out this probability directly is very difficult, but figuring out the opposite probability can be done using methods we already know.

The opposite statement:  If we have n people and none of them are born on February 29, what is the probability of none of them sharing a birthday?

Let's look at the problem step by step.

Only one person
Clearly, with just one person, we can't have two people sharing a birthday. Any of the 365 days this person is born on will mean he doesn't have a match, because there is no one to match with.

p(not sharing) = 365/365 = 1
p(at least two share) = 1 - 1 = 0


Two people
With two people, it possible but very unlikely for them to share. If we want to know about not sharing, the first person has 365 days that are acceptable but the second person has only 364.

numerator: 365 × 364
denominator: 365 × 365

Rounding to four places after the decimal

p(not sharing) = .9973
p(at least two share) = 1 - .9973 = .0027


Three people 
With three people, it possible but very unlikely for them to share. If we want to know about not sharing, the first person has 365 days that are acceptable but the second person has only 364, but the third person has 363 days on which he or she could be born.

numerator: 365 × 364 × 363 
denominator: 365 × 365 × 365

Rounding to four places after the decimal

p(not sharing) = .9918
p(at least two share) = 1 - .9918 = .0082

The probability of sharing is still small, but it's growing faster than you might expect. The formula for no shares among n people is the fraction

365 nPr n
365 ^ n

Many people are surprised how low n is when this fraction is less than 50%. If there are 23 people in a room - none of them with a Leap Day birthday - the probability that all of them have a unique birthday in the group is
 
365 nPr 23
365 ^ 23

which is .4927, rounded to four places. The odds of sharing has risen to .5073, or slightly over 50%