No - there is nothing especially unusual about this lottery outcome

 

Every so often we get a story like that today (from South Africa this time) about unusual lottery draws, with the implication that it is too incredible to believe this happened without some cheating going on.

The first thing to note is that the article does give the correct probability of this particular sequence being drawn on any one occasion, namely one in 42,375,200. This is calculated as follows:

• The probability the first ball drawn is 5 is 1/50 because there are 50 number in the main draw.
• The probability the second ball drawn is 6 (given that the first ball drawn is 5) is 1/49 because there are now 49 number left.
• The probability the third ball drawn is 7 is 1/48
• The probability the fourth ball drawn is 8 is 1/47
• The probability the fifth ball drawn is 9 is 1/46
• The probability the bonus ball drawn is 10 is 1/20 because there are only 20 bonus balls to choose from.
• So, the probability of getting the exact sequence 5,6,7,8,9,10 in that order is:

However, the particular order of the sequence does not matter, in the sense that 5,6,7,8,9,10 would still be the reported sequence even if the first 5 balls were chosen in a different order such as:

• 6,9,5,8,7, 10
• 9,8,6,7,5, 10
• 7, 9,6,8,5, 10
• etc

In fact, there are 5! (which is equal to 5 x 4 x 3 x 3 x 1) different possible permutations in which these numbers can be drawn. That is 120.

So, the probability of drawing the numbers 5,6,7,8,9 and bonus ball 10 is exactly:

Now this seems like an incredibly unlikely event. But, crucially, every possible set of winning numbers has exactly the same probability. Nobody would suspect anything if, say the numbers drawn were 3, 8, 21, 29, 47 and bonus ball 10. But the probability of getting that particular set of number is also one in 42,375,200.

What would be very unlikely is if, say, only one million lottery tickets had been sold and as many as 20 of these had all been for a winning sequence like 8, 21, 29, 47, 10. 

However, it is well known that most lottery players do not choose their numbers randomly. For example, numbers under 31 are far more commonly chosen than numbers over 31 because many people use birthdays of family members. That is why, when all the numbers in a winning draw are less than 32, there are far more winners than otherwise. Similarly, it is known that many people chose consecutive number sequences like 5,6,7,8,9,10. In fact, in the UK it is known that the most commonly chosen set of numbers is 1,2,3,4,5,6  (an average of 6,000 people chose this sequence for each draw in the early years of the UK lottery). So the fact that 20 people chose the consecutive sequence 5,6,7,8,9,10 is not at all unusual. In other words there is nothing at all to suggest the need for an 'investigation' as stated in the article.

But there is another important probability issue that most people completely misunderstand. While the probability of getting the particular numbers 5,6,7,8,9,10 in any one specific lottery draw is only one in 42,375,200, there are in fact 16 different ways we could get a consecutive sequence in any one draw (assuming the bonus ball must be at most 20), namely:

1,2,3,4,5,6

2,3,4,5,6,7

3,4,5,6,7,8

….

16,17,18,19,20

Any one of these sequences would have raised exactly the same concern. So, in any given draw, the probability of getting a consecutive sequence is actually one in 2,648,450 (not one in 42,375,200). While that is still a very small probability, we should also take account of how often lottery draws happen.

Let us assume there are 2 draws each week. Then, over a 10 year period there are 1040 such draws. The probability of getting a consecutive sequence at least once in a 10 year period is approximately 1 in 2547. But there are several hundred national and state lotteries in the world. If there were 300 such lotteries, then in any 10 year period the probability of getting a consecutive sequence at least once somewhere in the world is over 11.7% that’s a 1 in 9 chance**. So it really is not such an incredible 'coincidence' after all.

As explained in our book similar reasoning shows that real examples like:

• A woman in the USA wining the jackpot twice in 5 years
• A Bulgarian lottery in which the same numbers were drawn in two consecutive draws

are actually not particular unusual events (indeed the first is almost certain to happen somewhere in the world over any 10-year period). 

It's also interesting that this story seems to have grabbed the interest of academic statisticians far more than that of the US election - where there seems to be genuine statistical evidence of fraud.

Hat tip to Scott McLachlan for the story.

  **to be specific it is one minus the probability of getting NO consecutive sequence anywhere which is 1 -  (2546/2547) to the power of 300

 See also: 

• Can you improve your chances of winning the lottery even if you can afford just one ticket?
  
• The damaged winning lottery ticket story