September 5, 2019

This blog entry probably won’t teach you a lot about trading per se, but it may open up your mind to thinking outside of the box when it comes to probabilities.

This problem is known as the “Birthday Paradox”.  The premise starts with the question of how many people do you think it would take to survey, on average, to find two people who share the same birthday? Due to probability, sometimes an event is more likely to occur than we believe it to. In this case, if you survey a random group of just 23 people there is actually about a 50–50 chance that two of them will have the same birthday.  Sounds off right?  Let’s take a look.

First you would think that when you are in a room with 22 other people, if a person compares his or her birthday with the birthdays of the other people it would make for only 22 comparisons—only 22 chances for people to share the same birthday.

Don’t be so self-centered!  If the first person has 22 comparisons to make, but the second person was already compared to the first person, so there are only 21 comparisons to make. The third person then has 20 comparisons, the fourth person has 19 and so on. If you add up all possible comparisons (22 + 21 + 20 + 19 + … +1) the sum is 253 comparisons, or combinations. Consequently, each group of 23 people involves 253 comparisons, or 253 chances for matching birthdays.

When comparing probabilities with birthdays, it can be easier to look at the probability that people do not share a birthday. A person’s birthday is one out of 365 possibilities (excluding February 29 birthdays). The probability that a person does not have the same birthday as another person is 364 divided by 365 because there are 364 days that are not a person’s birthday. This means that any two people have a 364/365, or 99.726027 percent, chance of not matching birthdays.

In a group of 23 people, there are 253 comparisons, or combinations, that can be made. So, we’re not looking at just one comparison, but at 253 comparisons. Every one of the 253 combinations has the same odds, 99.726027 percent, of not being a match. If you multiply 99.726027 percent by 99.726027 253 times, or calculate (364/365)253, you’ll find there’s a 49.952 percent chance that all 253 comparisons contain no matches. Consequently, the odds that there is a birthday match in those 253 comparisons is 1 – 49.952 percent = 50.048 percent, or just over half!

Mind blown!

Source:  Scientific American

## Mike Shorr

Since 1994, Michael has been an on-the-floor market maker, Vice-President of Interest Rate Derivatives for Knight Financial Products and Director of Education and Options Instructor at Trading Advantage. He makes the oftentimes complex world of options and trading accessible to the novice and advanced trader alike. Michael has a Bachelor of Science degree in Statistics and Finance from the University of Illinois Champaign-Urbana. He presently is Director, Trader Education at ProsperTradingAcademy.

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