# Do winning numbers repeat?

2000-09-15 13:18

Vic Christie has observed that there have been quite a large number of repeated pairs in the winning numbers: after the first seventeen draws, no fewer than sixteen pairs had appeared more than once, with two of them even appearing three times.

This is at first glance surprising, since there are C(49,2)=1176 possible pairs and only C(6,2)=15 of them appear in any one draw. One might therefore think that it would take 1176/15=78.4 weeks before a pair is repeated.

What the above argument finds, however, is not the average time between repetitions, but the maximum possible time. It is impossible to have more than 78 consecutive draws without repeating a pair, since too many pairs have then been used up.

It is fairly easy to calculate what the chances are that a pair is repeated from one draw to the next. Using the table that appeared in my column "How many ways are there to play Lotto?", one finds that the chance that a ticket has at least two numbers right is 2111774/13983816, or about 0.151.

Now suppose that you never heard the proverb "lightning does not strike twice in the same place" and went out and bought a ticket with last week's winning numbers on it. Your chance of having at least two numbers right is 0.151, and that equals the probability that at least one pair from last week's draw has been repeated.

We can crudely approximate the chance that no pair from the previous k draws has been repeated by
 k 1-(1-0.151)

The actual chance is a bit lower because this approximation does not take into account that pairs get used up. For k=4 we get 0.52, a little over half, and thereafter it drops to under a half.

Therefore in any particular draw it is more likely than not that some pair from the previous five draws will appear again.

• Professor Laurie's weekly column on the Lotto appears every Friday. He is a mathematics professor at the Potchefstroom University. His special interest is numerical analysis.

Previous columns

• Is Lotto a game of skill? (5 May 2000)
• How many ways are there to play Lotto? (12 May 2000)
• Getting tips from a lotto result (19 May 2000)
• How big are the non-jackpot prizes? (26 May 2000)
• Is the jackpot big enough? (2 June 2000)
• Combining your chances (9 June 2000)
• Choose unpopular numbers (16 June 2000)
• Is the Lotto machine really fair? (23 June 2000)
• Playing in multiple draws (30 June 2000)
• Can Lotto make you rich? (8 July 2000)
• Your ticket's winning chances (13 July 2000)
• Two tickets in one draw vs one ticket in two draws (1 September 2000)