Five Paradoxes With Probabilities That Will Puzzle You
By: Marcel Moosbrugger
Tags: Probability, Paradox, Science, Technology

Five Paradoxes With Probabilities That Will Puzzle You

Photo from Andrea Piacquadio from Pexels

In our daily life, we constantly deal with uncertain situations. So — at least subconsciously — we are confronted with probabilities all the time. Moreover, it seems we have a good intuitive feeling for probabilities. Or do we really?

We know that flipping a coin 100 times will result in about 50 heads and 50 tails. Also, it’s clear to us that the probability of bumping into a friend on holidays abroad is very low, though not zero. However, there are situations when correct predictions following from probability theory are completely counter-intuitive.

Let’s dive into five paradoxes in probability theory and statistics that produce results that seem absurd but are true nonetheless.

1. Birthday Paradox

Image from Unsplash

You are at a friend’s party and there are 30 people present. What is the probability that two people have their birthday on the same day? What is your gut feeling? My first guess was that there is some chance but it should be somewhat low.

Now, the birthday paradox is about the following question:

How many people need to be at the party, so that we have at least a 50 percent chance of two people sharing their birthdays.

The paradox assumes that every day in the year is equally likely to be a birthday for a random person.

The surprising answer is 23! Only 23 people need to be present to have at least a 50 % chance of two people sharing their birthdays. For your friend’s birthday party with 30 people, the probability of two of them having the same birthday is actual over 70 %!

2. Simpson’s Paradox

Let’s assume we are business partners. We own five different coffee shops all over the city. As a marketing strategy, we give out vouchers to our most loyal customers in the hope they will return the favor and buy even more coffee.

However, we do not agree on how much the vouchers should be worth. To find out what’s the best choice, we hand out vouchers with different values. We collect data on how much profit we made with every loyal customer during the following month.

After one month everything seems to be clear: For every of our five coffee shops, we can see the trend: The more a voucher was worth the more profit we made with the corresponding customer.

However, when combining the data of all five shops together, the trend seems to be the complete opposite: The more a voucher was worth the less profit we made with a customer. How can this be?

A visual explanation of Simpson’s paradox. Image from Wikipedia.

This is exactly Simpson’s paradox:

A trend appearing in different groups of data can vanish when the data is combined.

3. Bertrand's Box Paradox

If you are familiar with the Montey Hall problem, this paradox is quite similar. In front of us there are three boxes:

Icon from Good Ware

One box contains two silver coins, one box contains two gold coins and one box contains a gold- and a silver coin. We do not know which coins are in which box. Now, we pick a random box and blindly draw a coin from our box. It’s a gold coin!

Now, the question is:

What’s the probability that the second coin in our box is also a gold coin?

My naive (and wrong) answer when encountering the problem for the first time was ½. I thought that because we drew a gold coin, our box is either the one with the two gold coins or the one with the mixed coins. In the first case, we would draw another gold coin and in the second case, we wouldn’t. Therefore, I presumed the probability should be ½.

The real probability is ⅔.

The reason for that is that the first gold coin we drew could either be the only gold coin in the mixed box, the first gold coin in the solely golden box, or the second gold coin in the solely golden box. And in two of these three possibilities, we will draw another gold coin.

Visual explanation of Bertrand’s box paradox. Image from Wikipedia.

4. Necktie Paradox

After the Christmas break, you are back at the office wearing your new tie you got as a present. Your colleague Bob also got a tie for Christmas. None of you know the price of your ties. You start arguing over who got the more expensive gift.

Bob and you agree to a bet: You are going to look up the ties’ prices. The one who got the more expensive tie has to give to the one with the cheaper one.

Two ties — Which one is more expensive? Icons from Freepik

Bob thinks that the bet is in his favor: His chances of winning/losing are 50/50. If he loses, he loses whatever his tie is worth. If he wins, however, he wins more than his tie’s value.

But from your perspective, you can do the exact same reasoning and conclude that the bet is in your favor.

But, the bet can’t be both in your and Bob’s favor. So where is the mistake?

To resolve the Necktie paradox we have to include the prices of the ties into the calculation: Let’s say one tie costs 100 € and the second one 50 €. If Bob wins, he wins a tie worth 100 €. If he loses, it means he owned the more expensive tie and just lost 100 €. So the possible gain and loss cancel and the bet is actually in no one’s favor.

5. Random Walks

This last fact is typically not listed as a paradox but I think it’s so fascinating and puzzling that it has to be included in this list.

Imagine an ant on an infinite piece of paper. After every second, the ant randomly walks into one of the four possible directions (forward, backward, left, or right). Every direction is equally likely. This scenario is what’s called a symmetric 2D-random walk. 2D, because the paper is two-dimensional.

We could also imagine the ant not being on a piece of paper but a string of fabric. In that case, the ant can only move forward or backward and we would be faced with a 1D-random walk.

Similarly, we can imagine a 3D-random walk where our “walker” is a drone or a bird which randomly moves into six directions (forward/backward, left/right, top/down).

Visualization of the 2D-random walk (left) and the 3D-random walk (right). Images by Vipul Lugade [1], [2].

Now the question is the following:

What’s the probability that a walker will return to its starting position?

The surprising answer is that for the ant on the fabric (1D) and the ant on the paper (2D) the probability is 1. But for the bird, there is a chance that it wanders off forever without returning. This is also true for dimensions 4,5,6…

It is mind-boggling to me that the walker always returns in dimensions 1 and 2 but the pattern breaks down from dimension 3 onward. I can look at the math but have to admit that on an intuitive level I still don’t fully understand it. The mathematician Shizuo Kakutani commented on this with a smile: