I was never extremely good at probability, but I got to teach a little bit of it in the past days. It wasn’t something extremely and I haven’t done any extremely special lessons, but they have opened my curiosity for searching for a couple of interesting problems. So, I thought I will write about 2 of my favorite problems/principals from probability theory.
1. Pigeonhole principle: this is more about counting, but hopefully you know how important counting is when it comes to probability. What is great about this principle is the fact that it is extremely straight forward and easy to understand. Also, I think this is one of the most used principles in day to day life without properly understand or realizing that we use it. The principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. This principle has great and fun examples that I like to read in my spare time. Some of the more popular problem are: hand shaking, sock picking and the birthday problem, which is the one I want to talk next.
2. Birthday problem: this is one of those fun and interesting talking point when it come to introducing the ways in which intuition leads you to strange answers when it come to probability theory. Basically, the problem or paradox states that in a set of n randomly chosen people, some pair of them will have the same birthday. This reminds me of my high-school when I had a classmate which was born in the same day as me, which was kind of strange. If we use the pigeonhole principle explained above, we can see that the probability reaches 100% (or 1) when we have a group of people which reaches 366 – 367 (since there are 365-366 days in a year). The interesting fact is that in fact this has a probability of 99.9% in groups of just 70 people and 50% (or 1/2) in groups of 23. Interesting, no? And unexpected I would say.
Reading through this I have discovered that this problem has an interesting application in cryptography, which immediately made me even more curios. It refers to birthday attach, which is used to abuse communication between two or more parties. The attack depends mostly on the likelihood of collisions found between attack attempts and a fixed degree of permutations, which takes us back to pigeonhole principle. I still need to read more about this, but I find it quite interesting.
Hope your enjoyed this small post and that it made you interested in the topic. Thank you for your help and support. Have a great day. You can find me on Facebook, Tumblr, Google+, Twitter, Instagram and WeHeartIt. I will try to post there as often as possible.
Don’t forget that maths is everywhere! Enjoy! ~LThMath