Lotteries and Strange Sequences

The numbers 5, 6, 7, 8, 9 and 10 came up in the South African PowerBall draw.

Unlikely events can happen. In South Africa, their PowerBall lottery saw the selection of balls numbered 5, 6, 7, 8 and 9. The PowerBall was 10.

This article looks at the chance of sequences in the South African PowerBall lottery.

What are the odds? (Image: Global News Canada)

Release those big money balls

Ithuba Holdings operates the national lottery in South Africa. There are several games, including two PowerBall draws. This is like the Thunderball in the UK national lottery games.

There are 50 balls (numbered 1 to 50) in the first machine. Of these 50, each draw selects five. There is a second machine which selects the PowerBall — numbered from 1 to 20.

Tickets that match all five balls plus the PowerBall win the jackpot.

In a fair machine, each possible combination has equal chance. The probability of each possible combination is 1 in 42.4m.

This is the probability for one ticket in the South African PowerBall.

How many consecutive numerical sequences are there? It depends on your definition. The PowerBall could be at the start, end, or part of a full numerical sequence. I write the sequence from 1 to 6 (1, 2, 3, 4, 5, 6) as 1–6.

  • Ending: There are 15 sequences which end with the PowerBall. Those sequences run from 1–6 (where 6 is the chosen PowerBall) to 15–20.
  • Starting or ending: Now, there are 35 consecutive sequences. We can start and end numerical sequences with the PowerBall. For the sequence 1–6, 1 or 6 could be the PowerBall. Also, there are five sequences with numbers more than 20 which can start with the PowerBall.
  • Involving: There are 105 consecutive sequences which involve the PowerBall. For sequences like 1–6, any of the six numbers could be the PowerBall. There are 15 possible sequences with numbers beyond 20 too.

We care about the combinations — not the order — of the chosen balls. This is why we multiply by 120 in the above calculation. For every five balls from the main pool, there are 120 ways to order that selection. There are about 42.4m unique combinations of possible numbers from this lottery.

As a result, the probability of a numerical sequence involving the PowerBall is around one in 404,000. By contrast, the probability of a numerical sequence ending with the PowerBall is about one in 2.8m. A calculation for ABC News appears to multiply this lower probability by 120 again. That is incorrect, as we are already dealing with combinations.

Calculating probabilities

The South African lottery has run for many years. The probability of interest is not that one machine produces a numerical sequence. It is the probability that a numerical sequence happens over those many years.

The other calculations are similar.

After 4,000 draws, the probability of PowerBall numerical sequences are:

  • Ending: 0.14%;
  • Starting or ending: 0.33%;
  • Involving: 0.99%.
Little by little, a little becomes a lot. (Image: R Pubs)

There is not a lone lottery. In South Africa, there are three Lotto games, two PowerBall draws, and more. In the UK, there are Lotto, Thunderball, HotPicks and Set for Life games. There are lotteries which cross national boundaries, like the Euromillions lottery.

With thousands of national lottery draws each year, some combinations become probable. A numerical sequence in a single draw is rare. When given enough opportunities, unusual events can be likely to occur.

The R code for the graph is available on R Pubs and GitHub.

This blog looks at the use of statistics in Britain and beyond. It is written by RSS Statistical Ambassador and Chartered Statistician @anthonybmasters.

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