This will be a companion piece to a previous one that explained how to use mathematical thinking to win every game of Monopoly. I’ve gotten several emails from people who used that article to win games of Monopoly against their kids, and really enjoyed seeing their kids get interested in math when they explained how they did it. This is a similar type of piece — something very cool that will help you or your kids connect math to real life. And if you plan the circumstances carefully, you can almost certainly win a bet with this one!
First, a brief roundup of things I’ve enjoyed reading and writing lately.
Recommended Reading
The inimitable
wrote something about the male/female gap in politics that I heartily recommend. This is the kind of link to keep handy for situations when what you desperately need is a link to send with the notation: “this explains my views better than I could, here you go”.Host of the Disaffected podcast (and one of my favorite humans)
identifies what you saw during the Trump/Harris debate that made you feel deja vu of a source that you couldn’t quite identify.Substacker and life coach
wrote something about the drama around “influencers”. I really appreciate when writers give me language for what I intuit but haven’t had the time and cognitive energy to put into words myself and she does a great job in this piece.ICYMI
I’ve had a prolific September, writing-wise. Here’s a brief description of things you might have missed.
I found a miracle cure (not hyperbole, folks!) for my attention span issues through a pair of trustworthy scientists that you’ve all heard of — Bret Weinstein and Heather Heying. Others are also finding that it’s a miracle cure for their attention problems, as the comment section attests, and I’ve gotten five emails from parents in awe of watching their teenagers become capable of focus, as if by magic. (Free for all readers.)
I did minor surgery on my friend
in my apartment. (Paywalled in my creative writing series, has a long preview.)Why the slippery slope is often not a fallacy, and why that matters. (Free for all readers.)
A personal story that sheds light on one of America’s most dire crises, the opioid nightmare. (Paywalled but has a long preview.)
I reviewed MATH-ish, the most recent book by Jo Boaler. She is the woman who screwed over your kids by rewriting the California math curriculum standards. As California is their biggest customer, the textbook publishers all conform to California standards, so this matters to everyone. (Paywalled but has a long preview.)
A personal update about the new job I’m starting and the changes it will mean for my Substack. (Free for all readers.)
The Birthday Paradox
Here’s a question to have fun with your kids around — especially if your kids have ever asked you when they’ll ever use math in “real life”.
Imagine you’re at a party. You look around the room and see people in various places, talking in small groups, dancing, or sharing snacks.
Now imagine that someone asks you, “What are the chances that two people here have the same birthday?”
(Throughout this essay, we’re referring only to the same birthday — month and day — not date of birth, which would mean they were born on the same month and day of the same year.)
It’s a great question, because the answer is very counter-intuitive.
If you had to bet your own money that two people at a party would have the same birthday (or two people in any group, party or not), how big would the group need to be before you’d be willing to bet?
Mathematically, what we are asking is how big of a group you need in order to make the bet one you’d be more likely to win than lose: the size of group that would get you over 50%.
The first time I heard this question, my gut instinct was that I’d want a very big group. At least 100 people, if real money was on the line.
Is your gut instinct that the number is pretty big?
Well, let’s figure it out!
The first person can have his or her birthday on any day. There are 365 possible birthdays.
The second person has a 1/365 chance of having the same birthday, and a 364/365 chance of having a different one.
Now there are two birthdays to possibly match, so the third person has a 2/365 of matching a birthday with someone and a 363/365 chance of having a different one from everyone else in the group.
The fourth person would have three birthdays to possibly match, so their chances become 3/365 of matching a birthday and 362/365 of having a different one from everyone else in the group.
It would go on like this, with each additional person’s chances of having a birthday that matches someone else in the group going up by 1/365 and their chances of having a different one going down by 1/365.
For something like this, that either will or will not happen, the chances must add up to 100%. An 80% chance that something will happen is by definition a 20% chance that it will not, for example.
I mention that because it’s usually easiest to calculate the odds of no match and subtract that from 1 (which equals 100%).
Just to make sure I don’t lose anyone — since saying “1 equals 100%” might seem a little confusing at first. Think of it this way. If you make a pie and you eat 25% of it and your children eat the other 75% of it, that one whole pie was split into two parts, a 25% part and a 75% part. 1 pie = 100% of the pie we’re talking about.
For this one event — the birthday comparison — there are two parts. The chances of a match, the chances of no match. And those chances must add up to 100%.
The chances of a matching birthday in a group of four people would be:
1 - [ (364/365) x (363/365) x (362/365) ]. That is, 1 minus the multiplied percentages of each new person’s chances of not matching. That gets us: 47,831,784 / 48,627,125.
Subtracting that from 1, we get .01635591. To find the percentage, move the decimal point two places to the right. A group of four has a 1.6% chance of a matching birthday.
For a group of ten people, it would be:
1 - [ (364/365) x (363/365) x (362/365) x (361/365) x (360/365) x (359/365) x (358/365) x (357/365) x (356/365)] = 11.6%.
For a group of fifteen people, it would be:
1 - [ (364/365) x (363/365) x (362/365) x (361/365) x (360/365) x (359/365) x (358/365) x (357/365) x (356/365) x (355/365) x (354/365) x (353/365) x (352/365) x (351/365)] = 25.2%.
Isn’t that a little surprising? There are 365 possible birthdays but you only need a group of 15 to have a 1 in 4 chance of a match!
Does that tell you that the number we are looking for — the number where it crosses over the 50% mark and makes you more likely to win this bet than lose it — is probably a lot smaller than you think?
The crossover point is……..
23 people.
That’s it! If there are 23 people at a party and you make this bet, you are more likely to win than lose. 23 people produces a 50.7% chance of a match.
And it only takes 41 people to get over the 90% point and 47 people to get over the 95% point, where it’s a VERY safe bet.
Why Does This Surprise Most of Us?
Our mathematical intuition is, in most cases, pretty bad. That’s why learning math is so important. Training our minds with the rigor and logic of mathematics improves our understanding of everything!
This problem surprises us because most of us think the problem is about comparing each person’s birthday to one specific day. We think of it in terms of “What’s the chance someone else at this party shares my birthday?”
But the real question is, "What’s the chance that any two people in the group share a birthday?" This makes a big difference because there are many possible pairs of people who could share a birthday.
In a group of 23 people, how many combinations are there?
To make this easier to understand, think of each person as a letter of the alphabet. 23 takes us from A to W, almost all the way to the end.
A can match to B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, or W.
B can match to A, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, or W.
C can match to A, B, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, or W.
To calculate how many possible pairs of 2 there are out of 23 people, we can use a known formula for combinations.
The known formula for combinations — which I will save, and explain in a future post — tells us that there are 253 possible combinations of two in a group of 23.
That’s why the birthday paradox works the way it does — because there are quite a large number of possible combinations here, and we only need one combination to be a hit.
And what I really like about this is that it sort of re-confirms the initial intuition — that out of 365 birthdays, you’d want a large group to make it a safe bet. A group that forms 253 combinations is a pretty decent size compared to 365 possible birthdays. We just have, until we learn math and train our minds, very bad mathematical intuition. So we get it on one level, the level of “we’d want a large group of combinations to make this bet”, but we miss it on the level of understanding that you don’t need that many people to set the conditions for that large group of combinations.
Most of us tend to have terrible intuition for how math works in real life, so we tend to make terrible guesses anytime good mathematical intuition would be helpful, and go through life that way.
That’s why learning math, and how to think mathematically, is a superpower for life. As this question shows, understanding math improves your thinking, period. The better you are at math, the more you will understand some of the intricacies of real life, and the better equipped you will be to make of your life what you wish.
Here is a google colab notebook to play around with this, if you like. Google will give you a scary warning, but there’s nothing in it except the code I wrote and that you see, and all it does it calculate this problem. It will be wonky on a phone, so use it on a desktop, laptop, or tablet.
Happy weekend, everyone!
The underlying maths has many other applications of course.
For example, suppose you're told that a facial recognition system has a 99% accuracy rate when comparing 2 head shots to decide whether those 2 head shots are the same person or not. We'll assume for simplicity that we have an equal rate of false positives (falsely saying it is the same person) and false negatives (falsely saying it is not the same person).
This sounds pretty good, and if you were to use it to ensure only the employees of a small company could enter its premises it would probably work well enough, at least if some allowance is made for the occasional mis-identification.
Now, we know that there's a 99/100 chance of correctly recording if 2 head shots are of the same person. But imagine using this to scan headshots taken (via CCTV) of members of the public as they walk down the streets of a busy city to check them against headshots of wanted criminals. Now the odds of getting things wrong become pertinent to your freedom to walk the streets unaccosted by the police.
How many comparisons would you need to do before the probability drops below 50%? The answer is 69. If you compare hundreds or thousands of people against the mugshots of wanted criminals you'll end up accusing lots of random civilians of being criminals and may well miss some of the wanted criminals along the way.
But suppose we raise the accuracy to 99.9%? The point at which the probability drops below 50% is 693. At 99.99% accuracy, the figure for dropping below 50% accuracy is 6,934. This application of facial recognition is going to need a very high accuracy to avoid lots of innocent people being flagged incorrectly as wanted criminals.
Now maybe there are ways around this by e.g. treating the initial match as a trigger for a deeper check of the person's identity to finally settle whether they're really the wanted criminal or not, but I'm sceptical of this kind of use of facial recognition because the maths indicate that an impressive level of pairwise-accuracy can nevertheless yield lots of error in such settings.
I was guessing over 365 to begin with. Lol. Interesting. Thanks for the shout out as well. :)