---
title: 'What Is the Probability of Winning the Lottery? | Mathematics Around Us | Boris Trushin'
source: 'https://youtube.com/watch?v=KX_KJ77dbJ4'
video_id: 'KX_KJ77dbJ4'
date: 2026-07-15
duration_sec: 563
---

# What Is the Probability of Winning the Lottery? | Mathematics Around Us | Boris Trushin

> Source: [What Is the Probability of Winning the Lottery? | Mathematics Around Us | Boris Trushin](https://youtube.com/watch?v=KX_KJ77dbJ4)

## Summary

In this video, math teacher Boris Trushin calculates the probability of winning the Russian lottery, explaining the combinatorial mathematics behind it. He contrasts the current rules with previous ones, showing how a small rule change drastically reduced the already minuscule odds.

### Key Points

- **Lottery Rules** [01:31] — Tickets have two fields with numbers 1 to 90. To win the jackpot, the first 15 numbers called must all be in one of the fields (top or bottom).
- **Total Outcomes** [03:37] — Number of ways to choose 15 balls from 90 is C(90,15) ≈ 45.5 quadrillion (4.55 × 10^16).
- **Current Winning Probability** [04:36] — Only 2 sets (top or bottom field) are winning, so probability = 2 / C(90,15) ≈ 1 in 23 quadrillion.
- **Global Perspective** [05:05] — If all 8 billion people bought a ticket, the chance that someone wins is about 1 in 3 million.
- **Previous Rules** [06:01] — Before, the 15 numbers could be anywhere on the ticket (both fields combined), so any 15 of the 30 numbers on the ticket were winning.
- **Previous Winning Probability** [07:34] — Number of winning sets = C(30,15) ≈ 155 million, so probability = 155 million / C(90,15) ≈ 1 in 300 million.
- **Comparison** [08:17] — The old probability was about 80 million times higher than the current one, though still extremely low.

### Conclusion

The current lottery rules make winning virtually impossible, with odds of 1 in 23 quadrillion. Even under the previous rules, the chance was only 1 in 300 million, highlighting how lotteries are designed to be nearly unwinnable.

## Transcript

One, two, three, four, five, everything seems to be fine. Now, back and not to be tell me off-camera? What the rule is that you need to introduce yourself. Yes, hello. My name is Boris Rushin. I'm an online math teacher. Some
time ago, some friends wrote to me that they were creating a new YouTube channel and wanted to make all sorts of interesting stories there. Their first story was about the lottery, about how it works, about the history of lotteries, about comparing lotteries in
different countries, and that's it. They asked me to give a short commentary related to the likelihood of winning the lottery in Russia. So, they released this video. There's only one video on this channel. I'll leave a link in the description, but they
cut out some parts from what I answered. And it seemed to me that you might be interested in this. It would be such a good story for the section "Mathematics Around Us." What's happening more difficult? What are the real probabilities of
winning these lotteries? So, enjoy watching. It will be especially enjoyable for those who are nostalgic for the old coward with long hair and a big beard. Tens of thousands of people saw her story and also wanted to win a billion. But  If
they had calculated their chances of winning, they simply wouldn't have stood in those lines. So I decided to calculate these chances for them and first studied the rules of the game. People stood in long lines for tickets that have two fields with
tickets that have two fields with numbers from 1 to 90 in them. To win the jackpot, the first 15 numbers that the presenter calls must be on your ticket, but not in any of the fields, namely in the upper or lower field. It seemed to me that the rules were
so simple that I could calculate this probability in 15 minutes. I suffered for a couple of hours and remembered why I was given 45 points on the Unified State Exam in mathematics, so I wrote to someone who will calculate everything. It's much faster than me. Well, look, let's begin.
How many different outcomes are there for 15 balls that can happen after pulling out 15 times? This is such fairly simple mathematics at the eighth grade level now, just in time for Anatoly, in the
eighth grade they are learning what needs to be calculated. Come on, the first ball can be chosen from any of 90 to any. Randomly, 90 can pop out.  so there are 90 options of what will end up in first
place after that no matter which ball we choose first we have 89 options of what can end up in second place Well, it turns out that for each of these 90 there are turns out that for each of these 90 there are 89 options, that is, choose two balls,
that's how many Well, continuing on, choose three balls, that's how many ways and so on 15th ball can be chosen 76 15th ball can be chosen 76 different remaining options Here But
this is not exactly what we counted, we counted How many ways to lay these 15 balls in a row Yes, the first, second, third, fourth and so on 15 and it doesn't matter to us We just need a set of 15 balls, it doesn't matter In what
order they appeared, so in fact, we counted each set many times How to count how many times we counted each set And this is exactly the same thing, yes Well, let's say we came up with a choice of some set of 15, let's see How many
different sequences can be obtained In the first position, you can put any of the 15 balls, the second, any of the 14 remaining ones and so on, therefore, in total, there are
The quantity is called the number of combinations of 90 by 15, it is designated like this. How many ways can you choose 15 out of 90 objects?
Well, it is clear that it is quite difficult to count by hand. But it turns out to be 45 46 845 quadrillion. That is, it turns out to be a quadrillion. That is, it turns out to be a 17-digit number. 4, there are 5-6, it does not matter.
17-digit number. 4, there are 5-6, it does not matter. 45.5. Let's write 46 by 10 16. That is, with good accuracy, this number is like this. It is simply very difficult to imagine 46 simply very difficult to imagine 46 quadrillion. So, that is how many
quadrillion. So, that is how many different outcomes there are out of 15 balls. And how many of them are suitable for us? In our ticket, there are two sets of 15 balls, which means that either this set must match this one. Or this one. That is, two suitable
this one. Or this one. That is, two suitable sets, which means the probability that our sets, which means the probability that our ticket will win is two divided by 46 by ticket will win is two divided by 46 by 10 16, that is, one divided by 2 and
10 16, that is, one divided by 2 and 3 by 10 16, one divided by 23 quadrillion. This is the probability that one will win.  a specific ticket to feel how insignificant this probability is if we assume
that all the people on the planet bought lottery tickets, that is, 8 billion tickets were bought, then the probability that any one of them will win is that any one of them will win is 8 billion times greater than this
seems 8 billion is a lot But if 8 billion is divided by 23 quadrillion, you get this: 8 by 10 9 divided by 2 and 3 by
8 by 10 9 divided by 2 and 3 by 10 16 This is approximately 1/3 multiplied by 10 6 That is 3 million one third The by 10 6 That is 3 million one third The millionth probability that someone will win
if everyone on the planet participates in the lottery is so small this probability turns out the chances of winning were not always so insignificant I dug around on the lottery website I found a text where they explain everything it is
found a text where they explain everything it is will explain everything to you in simple words In short, before, in order to win, the first 15 numbers that the presenters call had to be on your ticket and it does not matter whether it is in the upper or
lower field. Now these numbers must be  in one of the fields, either the top or bottom, sounds like a small change, but in fact, it changed all the chances of count it. Okay, look what’s changed. This
hasn’t changed. Yes, it was the same before, the number of different sets of 15 balls was still 46 trillion. So, what was different before? It was different before. The following was different. trillion. So, what was different before? It was different before. The following was different.
separate sets of 15 numbers, but one set of 30 numbers. Yes, all these 30 numbers are different, and we need at least 15 of them to match the set of 15 of them to match the set of balls that the presenter pulled out. So,
let’s see how many sets are suitable for us. It’s exactly the same problem. We have 30 numbers, and any 15 of them are suitable for us. This is what’s any 15 of them are suitable for us. This is what’s called out of 35. It’s calculated in exactly the same way:
called out of 35. It’s calculated in exactly the same way: 30 divided by 29, and so on by 16, divided by 30 divided by 29, and so on by 16, divided by 15, divided by 14, and so on by 1. If you 15, divided by 14, and so on by 1. If you count this carefully, you get 155
what was before, before, the probability of each specific set was the same, but before, 155 million sets were suitable for us,
that is, this card gave us the opportunity to win. Not like gave us the opportunity to win. Not like before, only two. Yes, but 155 million there are before, only two. Yes, but 155 million there are 155 million different sets of 15
numbers that are suitable for us to win with this ticket. So, it turns out that before, the probability was Well, how much, almost 80 million times greater. Well, you can calculate it. Yes. That is, before,
some kind of probability, you need to divide this again by 46 by 46 quadrillion. But this turns out to be approximately quadrillion. But this turns out to be approximately one divided by 300 million, that is, the
probability was also wildly small. One in 300 million that your ticket will win, but one in 300 million is still still 80 million times cooler than what it is now.
Well, that's it, such a short story from the series "Mathematics Around Us." Write. Are you interested in discussing such things? Leave your comments, likes, subscriptions, and that's it. See you next time. Bye, bye, guys.
But it will be normal that everyone will understand that you and I talked about all this in advance normal person in their right mind would calculate this. You just asked me, calculate this. You just asked me, I went and calculated, yes, okay, okay, we