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(db)
Now to find the P(db) we need to find the probability that nobody has the same birthday.
If there were 2 people to find the probability that they have different birthdays:
Person 1 - 365/365 (The number of days person 1 can have their birthday on)
Person 2 - 364/365 (Since we are finding the probability of different birthdays we have to make it 364 to exclude person 1’s birthday)
Thus the probability of them having different birthdays is
In this way if there were now 3 people:
Person 1 - 365/365
Person 2 - 364/365
Person 3 - 363/365
Thus the probability of having different birthdays is
The reason we continue this way is because they question asks the probability that “at least” 2 people have the same the birthday so we have to not only find the probability of that just 2 people but also for 3 people, 4 people and so on.
To find the P(db) for 30 people it would be:
This can be typed into your calculator manually however it will be a long process so instead we use factorials. When using a factorial function we multiply a series of descending natural numbers. The function is represented by the “!” symbol.
5! = 5 × 4 × 3 × 2 × 1 = 120
When you divide factorials you get:
As seen above the 4 × 3...