This week the BBC, The Guardian, and ABC published a story about a suspicious consecutive streak of winning numbers in the South African national lottery.
"South Africa’s lottery probed as 5, 6, 7, 8, 9 and 10 drawn and 20 win."
Gasp! Horror! The scandal!
Have the good people of South Africa been cheated? Or is it a case of the high probability of low-probability events*?

The Lottery
The BBC article tells us a little bit about the South African Lottery under question.
You pick 5 balls out of 50 possibilities and then one power ball out of 20.
According to the BBC, there’s a one in 42,375,200
How do we calculate that?
There are two parts to the draw. Picking the first 5 numbers, and then choosing the power ball.

The First Five
In this problem, we have to pick 5 numbers out of 50, with the number of remaining possibilities reducing each time a number is chosen. The order the numbers are drawn doesn’t matter. So, a drawing of 5,6,7,8,9 is the same as 8,5,9,7,6 (which was the actual drawing). This problem is called a combination.
The first number we draw is out of 50 numbers. We are choosing 5 numbers, and we don’t care about the order. So, we could choose any of our 5 numbers for this draw.
49 possible numbers are left for the second draw, with 4 numbers for us to select.
For the third draw, our choices are 3 out of 48, the fourth draw 2 out of 47, and the fifth draw 1 out of 46.
The number of outcomes is the possible numbers of each draw multiplied together, divided by our choices in each draw multiplied together.
or (50 x 49 x 48 x 47 x 46) / (5 x 4 x 3 x2 x1)
= 254,251,200 / 120
= 2,118,760
Combinations
Another way to look at this problem is with the combination formula. (This is the only mathematical formula in this blog … I promise)

n! means n factorial, which is n multiplied by every number smaller than it, all the way back to 1. For example, 4! is 4x3x2x1 or 24.
To model the lottery draw with 50 possibilities (n) and 5 choices (k), n-k =45.
50! / 45! x 5!
The first 45 terms of 50! are canceled by 45! And just like above, we have:
(50 x 49 x 48 x 47 x 46) / (5 x 4 x 3 x2 x1)
Some Code
In python, this is:
import math
def combinations(n,k):
numerator = math.factorial(n)
denominator = math.factorial(k) * math.factorial(n-k)
return numerator / denominator
combinations(50,5)
You can even model it in excel.
=COMBIN(50,5)

The Power Ball
This part is easy; the power ball is 1 number out of 20. So we multiply the first five by 20 to get our overall possibilities:
20 x 2,118,760 = 42,375,200
Just like the BBC said, there is a 1 in 42 million chance of drawing 5,6,7,8,9 and a power ball of 10. With these odds, not to mention 20 winners, this does look suspicious indeed.
Suspicious or Not
Why does it have to be a consecutive streak of 5,6,7,8,9,10, what about 1,2,3,4,5,6 or 30,31,32,33,34,35? Wouldn’t these be newsworthy as well?
There are 46 possible streaks of 5 numbers between 1 and 50. The largest number can be no more than 50, so our last possible streak is 46,47,48,49,50. However, our power ball number only goes up to 20
(That 30–35 streak really would be newsworthy. It features a power ball that doesn’t exist)
A power ball of no more than 20 restricts us to 15 consecutive streaks with the power ball as the last number – ending at 15,16,17,18,19,20pb. The power ball ranges from 6 to 20.
Let’s also include streaks with the power ball at the beginning of the streak. There are 20 of these streaks, ending at 20pb,21,22,23,24,25.
That is a total of 35 streaks. 35 out of 42 million. Still not great odds.
Place and Time
What about other lotteries? The chance of a streak in one specific lottery is remote. But how about the chance of a streak in any lottery anywhere?
According to this estimate, there are about 180 different national or state lotteries across the world. Many of these lotteries have weekly draws; some have bi-weekly draws. Let’s assume that half are weekly, and half are bi-weekly for an average of 1.5 draws per week.
For simplicity, let’s also assume that the Probability of winning the South Africa lottery is roughly the average of all international draws. This assumption looks reasonable in comparison with California (1:41M) and UK (1:45M). But not at all reasonable compared with the absurdly long odds of EuroMillions (1:140M), PowerBall (1:292M), and MegaMillions (1:303M). Fortunately, some state lotteries bring things back down to earth – a couple of examples: Arizona (1:7M), Maine (1:4.5M).

How Rare is a Streak?
Let work out the number of possible consecutive streaks, across the world, in a year
35 streaks x 180 lotteries x 1.5 per week x 52 weeks
= 491,400 streaks
Divide the number of South African lottery combinations by the number of streaks
42,375,200 / 491,400 = 86.23
This means that if our assumptions are roughly correct, we could expect to see a streak in a major lottery once every 86 years. Not all that often, but not completely out of the ordinary either. There are plenty of 1 in 100-year events that we experience. Usually, these are disasters for some reason – floods, fires, droughts, and this wretched pandemic.
Once in 86 years doesn’t look nearly as off as one in 42,000,000. So, breathe easy, South Africa; you are probably not the victim of sophisticated lottery fraud.
In addition, making some changes to our assumptions could boost the probability of consecutive streaks. For example, we could include smaller lottery draws that were not in our original list, increase our power ball possibilities to greater than 20, or include streaks with the power ball in the middle of the sequence. Before you know it, we could get down to a streak every 5 or 10 years.
What About the Multiple Winners?
This is the part of the lottery that’s not random. People look for patterns or ways to find meaning out of chaos. A streak of numbers is far more appealing than any old random sequence. People will select personal "lucky" numbers, or birthdays, or a streak, or whatever else brings them meaning. That’s how you get 20 winners.
What Does This Mean?
As of writing, both Powerball and MegaMillions are approaching $300M in prize money. This is large enough to attract attention but not yet large enough for lottery frenzy. (Check out the billion-dollar peaks in January 2016 and October 2018).

Should you play the lottery? No, you absolutely should not. But if you do … get a ticket for the entertainment value only. Imagine those cars, that travel, and your own private island. Rent the dream for a few days until the cold hard hand of probability snatches it away, along with your two dollars.
And when you pick your numbers, don’t choose like most people. Pick numbers so that you’re less likely to have to split your winnings. Avoid "lucky number 7", don’t pick birthdays, and whatever you do … don’t choose a consecutive streak.
Thanks for reading. Good luck!

References
"High probability of low-probability events" – This catchy phrase originated in Triggers by Marshall Goldsmith.
Draw Update
Since I first wrote this story, both PowerBall and MegaMillions exploded in size and both draws have a winner
The PowerBall jackpot of $731.1 Million, drawn on January 20, went to a single winning ticket sold in Maryland. Winning numbers were:
40, 53, 60, 68, 69, plus Powerball 22
Mega Millions was won on January 22, with a final jackpot value of $1.05 billion, to a single ticket sold in Wisconsin. Winning numbers were:
4, 26, 42, 50, 60, plus Mega Ball 24