“I was never very good at math,” “Oh, I hate math,” “Why?” I almost always get similar responses from people when I tell them I am a math major. I’ve begun to really notice how standoffish most people are toward basic math and statistics and how detrimental that fear is in terms of our basic reasoning skills. Very often, I’m amazed by how easily some news stations and some companies (like all the diet pill commercials) are able to blatantly lie to the average consumer with statistics. I think that everyone needs to have a fundamental understanding of statistics and math to better understand the way the world works, so that’s what I hope to accomplish with this series: I want to help make you all more comfortable with math.
For a basic example, let’s discuss a basic probability question, the “Birthday Paradox” which is reasonably well-known. Let’s say there are 35 people in my math class and want to see what the probability is that any two members of my class share a birthday. The probability seems pretty low at first seeing as there are 365 days in a year, but its actually 81.44%! Let’s figure out why this is the case…
The first thing to consider here is that obviously, the probability reaches 100% when the class size reaches 366 (by the pigeonhole principal) because there are only 365 independent and equally likely (1/365 probability of occurring) birthdays (we’re excluding February 29th because it is so rare).
Let’s consider things from a different perspective, instead of thinking about everything as the likelihood that two people share a birthday, think about the likelihood that they do not.
For a 1 person class there is a 365/365= 100% chance that there is no one that shares the birthday.
So, for a 2 person sized class, there is a (365/365)*(364/365)= 99.73% chance that the two people not share the same birthday.
For a 3 person class, there is a (365/365)*(364/365)*(363/365)= 99.18% chance that two people do not share a birthday.
Continuing this, we can generalize that for any n person class size, there is a ((365*364*363*362*…*(365-n+1))/(365^n)) chance that no two people share a birthday.
Using this formula for my class of 35 kids, we can arrive at the conclusion that my class has roughly an 18.56% probability of no one sharing a birthday. This means that there is a (1-.1856) = 81.44% probability that two people share a birthday!
I hope that basic example was a helpful tool for some of you to at least see how misleading “common sense” can be when dealing with numbers and how changing the way we look at things can make all the difference! Next time I’ll be discussing the Monty Hall problem.