South African Powerball Comes Up 5, 6, 7, 8, 9, 10. What Are The Odds? Let’s Find Out!

Was the lottery rigged? Let’s do the math!

10 min readDec 11, 2020

Screencap of South African Lottery results taken December 2, 2020

On December 1, twenty lucky people won the PowerBall lottery in South Africa, and are millionaires (in rand) . They will each receive about $371,630 if converted to equivalent U.S. currency. But that’s not the insane news. The wildest development is that the numbers selected, though not in order, were 5, 6, 7, 8, and 9. The bonus Powerball number selected was, surprisingly, 10.

According to the South African lottery officials, all 20 winners played manual numbers — meaning a computer did not select their numbers. As expected, eyebrows are being raised along with calls for inquiries. “I’m calling for Justice here,” said one tweet.

So just how rare is a lottery pull of six consecutive numbers? The answer might surprise you. Before we start, let’s review the South African PowerBall rules. According to the website, you use a pen or pencil to mark five numbers from a possible set of one through fifty. Then, for an extra dollar or so (I haven’t done the U.S. dollar conversion), you can opt for the “plus” option, the PowerBall. That’s one extra ball numbered from one to twenty.

To determine the odds of a consecutive selection, you need to use a combination function to determine all possible outcomes. The combination function uses factorials where n is the total set; in this case, the number of balls: 50. The next factor is m, which is the number of balls we must select without replacement. For the South African Powerball, that’s five balls.

Replacing the placeholders for five balls and then using the COMBIN() function in Excel, we find a total of 2,118,760 total possible outcomes. Excel is awesome and saves us from messy factorial multiplication.

Now we need to determine the set of successful consecutive outcomes. Those could be numbered one through five, three through seven, or any combination of five numbers. We use a permutation function since we don’t care about the order of the balls selected. Remember, we can choose the balls out of order, just as in the lottery yesterday; the only thing we care about is that all five balls are consecutive when we are done and lay them out on the table.

If we start with an array of one through five, the permutation could be 12345, 13245, 54123, and so on. The permutation formula shows us that there are 120 possible combinations for the numbers one through five alone. But we could also have a permutation that begins with two and ends with six; there are 120 combinations of those too. To arrive at the total number of possible consecutive five-number outcomes, we multiply 120 by 19.

We multiply by 19 because the Powerball we select last is only numbered one through 20. Remember, that’s in a separate pool. Since we can never choose a Powerball numbered 38, for example, we cannot include an array of five numbers that might be inclusive of 38 or any number above 20. Additionally, we stop at 19 rather than 20 because we have to leave “room” for a pick from the Powerball. That number might be 20, or it could be 11 — it doesn’t matter. We just know that it needs to fit in our final array of 6 numbers selected.

Now that we know the total possible number of selected outcomes, we have a simple probability formula.

That’s not one percent, that’s point one percent, or one-tenth of one percent. It’s an extremely low number, but we still aren’t done. We need to select that Powerball, and it needs to be consecutive. Simple probability will help here. Remember, this is a separate pool of balls, numbered from one to twenty. Selecting a single ball would be represented as the number of desired outcomes divided by the number of possible outcomes, the same as the final formula. That simple math results in a 5 percent chance of selecting the number 10 or any other number from the pool.

For the final result, we need the product of the two probabilities combined, similar to determining the outcome of two independently rolled dice. So, we take P(A) — the likelihood of five consecutive numbers and multiply by P(B) — the final PowerBall probability.

And that’s the final result of a computer randomly selecting six consecutive numbers in the South African Powerball lottery: 0.0054%. In other words, the lottery system has a 1 in 18,586 chance of selecting five consecutive numbers and then drawing a 6th in sequence for the Powerball.

To put those long odds in perspective, if you live to be 80 years old, you have a greater chance of being killed by lightning in your lifetime, which is a 1 in 15,300 chance. You also have a greater possibility, 1 in about 8000, of death by sunstroke.

The odds of holding the winning ticket

But those are just the chances of the lottery system randomly selecting any six balls that happen to be consecutive. What about the people that won the top prize? What were their odds of choosing the exact same numbers of five through ten last night? If we do the math, the decimal places become insane. Instead, let’s stick with odds for this one.

Selecting the same five numbers, regardless of order, is about 1 in 2.11 million. Adding the sixth PowerBall bonus number would be about 1 in 42.3 million. Sure, it’s possible. Twenty lucky people just found out.

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This article is for informational purposes only. It should not be considered Financial or Legal Advice. Consult a financial professional before making any major financial decisions.

South African Powerball Comes Up 5, 6, 7, 8, 9, 10. What Are The Odds? Let’s Find Out!

Was the lottery rigged? Let’s do the math!

The odds of holding the winning ticket

Written by Udashingmahmoodm