If I were to wager that at least two people in my classes do not share a common birthday, I would ‘almost surely’ be wrong. In my class of 28 students, there is about a 70% chance of two people sharing the same birthday.
Let’s say I am to ask everyone their birthday. The second person I ask has a 365 out of 366 chance that they have a different birthday from the first person. That is about a 99.7% certainty that their birthday does not match. The third person also has a pretty good chance that his or her birthday does not match either one of the first two people (364 out of 366 chance). The 28th person has a 338 out of 366 chance of not sharing anyone else’s birthday (still a 92% chance of not having a matching birthday). Thus, each individual’s chance of missing everyone’s else birthday is pretty good. However, we want to know the likelihood of no one having matching birthdays. With all those chances combined, we actually have a very good chance of two people sharing the same birthday. To compute the likelihood that all of my students do not share a birthday, we can first determine the likelihood that two people share the same birthday. To calculate, we multiply together all the individual chances: 365/366 x 364/366 x 363/366 … x 338/366. In that case, we get a number around .345. Thus, we only have a 35% chance of two people not sharing the same birthday. That give me a 65% chance of being correct if I guess that two people share the same birthday.
This example is quite interesting and fun to demonstrate at parties (I no longer get invited to fun parties). However, there is a much better application of these ideas. One example is applying for a job or internship. Let’s imagine that I am really an awful candidate and that I don’t fit well at any position. Maybe I have a 1 in a 100 chance of getting a job. Let’s say there are 50 places at which I have that same chance of getting a job. Amazingly I have about a 40% chance at getting at least 1 job if I apply to all 50 places. The first job I apply to I have a 1 in a 100 chance. The second one I also have that same chance. Now, if I multiple 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100 x 1/100, I get the probability that I won’t find a job. If I subtract that number from 1, I get the probability of finding a job. Thus, I have above a 39.4% chance of getting at least one of those jobs.
The simplest formula for these problems is: 1-p^n, where p is the probability of finding a job and n is the number of jobs of which I applied.