Math IA - The Birthday Paradox
“What is the probability that at least 2 people in a room of 30 random people will have the same birthday?”
Probability is always surrounding us from stock markets to the ever-simple heads or tails. This very complicated area of mathematics can be explained in a simpler way. It is how likely an event is to happen. The probability of an event will always be between 0 and 1. The closer it is to one, the more likely the event is to happen.
I chose this topic because when I first read the birthday problem in the textbook, I tried to solve it repeatedly but each time I would get a very low probability. After re reading the question for the 20th time I finally realized my error. I was considering the probability that people would have the same birthday as me when the question was focusing on the probability that anyone in the room has the same birthday. When I finally managed to solve the question I didn’t know what else this could be used for so I did more research. That is when I discovered that the birthday paradox can also be used to crack hashing algorithms and can be used in cryptography.
The square below represents the sample space (all possible outcomes)
We have not calculated the probability yet so lets assume that the white represents the number of people who have the same birthday. We will call this P(sb).
The gray area represents the number of people who have different birthdays. We will call this P(db).
The reason we use this formula in particular is because the birthdays of 30 random people in a room are all independent events that do not rely on one another for it’s occurrence.
Since the maximum probability of an event is 1 this means:
P(db) + P(sb) = 1
This in turn means:
P(sb) = 1 - P...
... middle of paper ...
...tal part of lives just like privacy. Using cryptology provides mechanisms through a digital signature. This signature is inserted using a key (that only the writer of the email possesses) whilst a timestamp binds itself to the document. This type of cryptography is used to control access of security installations or pay-per-view television channels.
This IA started out with me trying to solve a question that I had tried to solve for ages and now I have realized that we ignore some amazing mathematics because we are so self-involved. The real birthday problem is not the question itself but it is a test of our ignorance. My research and this problem solving have helped be more objective with my thinking and understand how amazing probability really is.
I definitely want to solve more of these types of questions and learn more about the other fields of mathematics.
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