Real Mathematics – What are the chances?! #3

Thinking of a number

I am thinking of a number from 1 to 10. (It is 6.)

You have to guess this number.

You have only one shot which means that your success rate is 1 in 10. In other words it is 10%.

Let’s reverse this train of thought and only consider rate of failure which is 90%. (100-10=90)

Q: Assume that we have a group of five people. We ask everyone to write down a number from 1 to 10 simultaneously. What is the probability of having at least 2 people writing down the same number?

First answer comes to your mind is 50%, isn’t it? Because 5 out of 10 makes 50%. Although things may not be as they seem in probability theory.

We’ve already talked about this case for two people. Probability was 9/10 or 90%. How about for three people?

For the first person, there are 10 available numbers from 1 to 10: 10/10.

For the second one, there are 9 available numbers from 1 to 10: 9/10.

For the third person, there are 8 available numbers from 1 to 10: 8/10.

Probability of these three cases to happen can be calculated with multiplication of them: (10/10)*(9/10)*(8/10) = 0,72. This means that %72 of the time three people will write down different numbers.

For four people probability can be calculated like this: (10/10)*(9/10)*(8/10)*(7/10) = 0,504. This means that 50,4% of the time four people will write down different numbers.

For five people probability will be: (10/10)*(9/10)*(8/10)*(7/10)*(6/10) = 0,3024. This means that 30% of the time five people will write down different numbers from 1 to 10.

Probability is way above 50%. Are you surprised?

Birthday Paradox

Whenever you meet someone new, the day you were born will be one of the subjects of this first conversation. If there is at least one female in that conversation your horoscope signs will definitely be mentioned. Now think for a second: Have you ever met someone who shared the same birthday with you?

You probably haven’t thought about it since you know that it is a highly unlikely thing. Because if you consider one year as 365 days (sorry 29^th of February)this probability is 1 over 365 which makes 0,27%.

Q: How many people do you need in a group to have 50% chance that two of them will have the same birthday?

At first you might think that you would need 365/2 people for that. But this is not the true answer.

This question can be answered with the same logic we used for the thinking of a number game. And when you apply the same method, you will end up with a bizarre result: Only 23 people are needed for such probability!

Let’s start analyzing this result slowly as we did in the thinking of a number game.

For the first person there will be 365 available days out of 365 total days.

For the second person there will be 364 available days out of 365 total days.

For the third person there will be 363 available days out of 365 total days.

…

For three people, not having same birthday is 99,2%.

For four people it is 98,4%.

For ten people it is 88,3%. As we continue calculating we can see for 23 people it is 49,3%.

This means that 50,7% of the time two people will have the same birthday among a group of 23 people.

This is an astonishing result!

One wonders…

Determine how many people do you need in a group such that 99% of the time you will be able to find two people who will have the same birthdays?

M. Serkan Kalaycıoğlu