Lottery/Powerball math question

[TofuFighter]

So... in South Africa yesterday, the Powerball results were interesting, and when they are this interesting, a lot of people begin to question whether there are some shenanigans going on.

The powerball here is structured as follows:
Select 5 balls from a pool of 45
Select 1 powerball from a pool of 20

The results of the powerball were 5 / 6 / 7 / 8 / 9 and Powerball 10
There were 20 winners with that combination

Results page: here
Lottery card: here

What are the odds of this exact sequence of numbers being drawn?
I guess it's impossible to calculate the odds of that number of winners without knowing the number of entries.

Is this sequence necessarily unusual? Unlikely, naturally, but probability being what it is just means that any sequence is possible.
I understand that the system of drawing numbers is electronic and thus there is some scope for manipulation, but at the same time, if one were to manipulate the results, a sequence like this would seem like exactly the wrong way to go about it. What other arguments are there against the conspiratorial thinking that is likely to arise?

I don't play any kind of lottery games so I'm not invested in this, but hearty thanks to anyone who takes the time to answer or comment.

 

[CORed]

So... in South Africa yesterday, the Powerball results were interesting, and when they are this interesting, a lot of people begin to question whether there are some shenanigans going on.

The powerball here is structured as follows:
Select 5 balls from a pool of 45
Select 1 powerball from a pool of 20

The results of the powerball were 5 / 6 / 7 / 8 / 9 and Powerball 10
There were 20 winners with that combination

Results page: here
Lottery card: here

What are the odds of this exact sequence of numbers being drawn?
I guess it's impossible to calculate the odds of that number of winners without knowing the number of entries.

Is this sequence necessarily unusual? Unlikely, naturally, but probability being what it is just means that any sequence is possible.
I understand that the system of drawing numbers is electronic and thus there is some scope for manipulation, but at the same time, if one were to manipulate the results, a sequence like this would seem like exactly the wrong way to go about it. What other arguments are there against the conspiratorial thinking that is likely to arise?

I don't play any kind of lottery games so I'm not invested in this, but hearty thanks to anyone who takes the time to answer or comment.

I'm not sure of the exact number, but the odds of that particular combination coming up are the same as the odds that any other single combination will come up, which is probably 10's of millions to one. The reason a lot of people had that combination is that people who pick their own numbers rather than a randomly selected set often tend to pick sequential numbers, birth dates, etc. If one were going to rig a lottery, they would be wise not to pick a combination that a lot of other people selected, so they don't have to split the prize.

 

[tusenfem]

The odds are easy to calculate

(5/45) * (4/44) * (3/43) * (2/42) * (1/41) * (1/20) = 120/2932221600 ~ 4E-8

or about 1 in 25 million

 

[The Atheist]

I don't play any kind of lottery games so I'm not invested in this, but hearty thanks to anyone who takes the time to answer or comment.

The maths is as given - but I'd be interested to know if that's more winners than usual?

One interesting thing we found in NZ about random numbers is that more people win Lotto than maths tells you you'll get. The originators expected a jackpot every second week on average, but they only happen about once every ten weeks.

 

[Lothian]

What are the odds of this exact sequence of numbers being drawn?

tusenfem has, i think provided the odds assuming the numbers were drawn in the order you listed. If the numbers could be drawn in any order the odds are 120 times less or about 1 in 200,000 (again I think, a while since I did any probability)

I guess it's impossible to calculate the odds of that number of winners without knowing the number of entries.

Yes, but as TA suggests people don't pick numbers as randomly as they are drawn so some combinations will have lots of winners others less so. Draws with numbers over 31 tend to have fewer winners as many people use birthdays to pick numbers

 

[zooterkin]

tusenfem has, i think provided the odds assuming the numbers were drawn in the order you listed. If the numbers could be drawn in any order the odds are 120 times less or about 1 in 200,000 (again I think, a while since I did any probability)

Yes, but as TA suggests people don't pick numbers as randomly as they are drawn so some combinations will have lots of winners others less so. Draws with numbers over 31 tend to have fewer winners as many people use birthdays to pick numbers

Indeed. There is no way to increase your chances of winning the lottery (apart from buying more tickets), but, if you must play, you can increase the chances of being the sole winner by picking a combination that no-one else has chosen. The frequency of chosen numbers is often available; as Lothian says, birthdays are very often used. Some lotteries even have the option for a random number selection which, I think, avoids combinations already picked.

 

[TofuFighter]

tusenfem has, i think provided the odds assuming the numbers were drawn in the order you listed. If the numbers could be drawn in any order the odds are 120 times less or about 1 in 200,000 (again I think, a while since I did any probability)

I think maybe my OP wasn't quite clear on the question. I didn't mean that sequence in that order. I think tusenfem (which I thank him for) is correct that regardless of the order, the odds are around 1 in 24mil. The reason i think that is only because I did a bunch of google-calculating before creating this thread and that was the answer I kept getting to.

I think the odds become astronomically smaller of drawing them in that particular order, but i'll qualify that by adding that it's been ages since I did any meaningful probability calcs as either.

 

[TofuFighter]

The maths is as given - but I'd be interested to know if that's more winners than usual?

One interesting thing we found in NZ about random numbers is that more people win Lotto than maths tells you you'll get. The originators expected a jackpot every second week on average, but they only happen about once every ten weeks.

Results page here

This particular jackpot was R113m, carried over for 2 months since the start of October, so zero winners for the previous 17 draws. then 20 winners in this draw.

summary

This jackpot was R113m, which had 20 winners, carried over 17 draws
Previous jackpot was R121m which had 1 winner, carried over 21 draws.
Previous jackpot was R31m which had 1 winner, carried over 5 draws
Previous jackpot was R77m which had 1 winner, carried over 18 draws

I haven't gone further back.

Considering that immediately prior to this event there had been close to 60 draws with 3 winners, it does seem a bit unusual that there would be 20 in this one.

mitigating:
- there are more entries with higher value jackpot
- there may have been more winners exactly because of the unusual number sequence

 

[Dave Rogers]

I think the odds become astronomically smaller of drawing them in that particular order, but i'll qualify that by adding that it's been ages since I did any meaningful probability calcs as either.

It's worth pointing out, though, that the odds are nevertheless no smaller than those of drawing any other chosen specific sequence in a chosen specific order.

Dave

 

[TofuFighter]

It's worth pointing out, though, that the odds are nevertheless no smaller than those of drawing any other chosen specific sequence in a chosen specific order.

Dave

Thank you. CORed also pointed this (or a similar concept) out.
When i was mulling it over, prior to posting the OP, I had considered that but sometimes when i am lazy my mind gets fuzzy about the logic. glad to have it noted.

 

[lomiller]

So... in South Africa yesterday, the Powerball results were interesting, and when they are this interesting, a lot of people begin to question whether there are some shenanigans going on.

The powerball here is structured as follows:
Select 5 balls from a pool of 45
Select 1 powerball from a pool of 20

The results of the powerball were 5 / 6 / 7 / 8 / 9 and Powerball 10
There were 20 winners with that combination

Results page: here
Lottery card: here

What are the odds of this exact sequence of numbers being drawn?
I guess it's impossible to calculate the odds of that number of winners without knowing the number of entries.

Is this sequence necessarily unusual? Unlikely, naturally, but probability being what it is just means that any sequence is possible.
I understand that the system of drawing numbers is electronic and thus there is some scope for manipulation, but at the same time, if one were to manipulate the results, a sequence like this would seem like exactly the wrong way to go about it. What other arguments are there against the conspiratorial thinking that is likely to arise?

I don't play any kind of lottery games so I'm not invested in this, but hearty thanks to anyone who takes the time to answer or comment.

IMO this isn't a math question but a psychology question. The odds are random but human behavior is not necessarily random. Most people avoid sequences like that under the mistaken belief that it's somehow less likely than other combinations, however I don't find it surprising that there are at least some people attracted to it. Some may think "no one else would pick this" while others choose it because it's "easy" or "lucky".

Overall there are going to be combinations that people puck much more often than random chance would dictate and the winning numbers just hit one of them. 1, 2, 3, 4, 5 and Powerball 6 probabaly would have had even more winners.

 

[Ziggurat]

I think maybe my OP wasn't quite clear on the question. I didn't mean that sequence in that order.

Let me ask a question, since I'm not very familiar with powerball. Does the order even matter?

What I mean by this is not just whether you can win with matching numbers in a different order. If the order doesn't matter to whether you win, then it doesn't matter to the order that they're drawn in. And in that case, they don't even have to announce the numbers in the order that they're drawn, they could just list them in ascending order every time. So is that how it's done? Do they list it in ascending order, or in draw order?

 

[lomiller]

Results page here

This particular jackpot was R113m, carried over for 2 months since the start of October, so zero winners for the previous 17 draws. then 20 winners in this draw.

summary

This jackpot was R113m, which had 20 winners, carried over 17 draws
Previous jackpot was R121m which had 1 winner, carried over 21 draws.
Previous jackpot was R31m which had 1 winner, carried over 5 draws
Previous jackpot was R77m which had 1 winner, carried over 18 draws

I haven't gone further back.

Considering that immediately prior to this event there had been close to 60 draws with 3 winners, it does seem a bit unusual that there would be 20 in this one.

mitigating:
- there are more entries with higher value jackpot
- there may have been more winners exactly because of the unusual number sequence

Again, this is entirely possible when people select their own numbers. Some numbers are selected more often then others and some numbers are selected together more frequently than random chance would dictate. The random draw will either hit or miss one of these underlying patterns resulting in more winners if it hits one and fewer winners if it does not.

 

[lomiller]

Let me ask a question, since I'm not very familiar with powerball. Does the order even matter?

What I mean by this is not just whether you can win with matching numbers in a different order. If the order doesn't matter to whether you win, then it doesn't matter to the order that they're drawn in. And in that case, they don't even have to announce the numbers in the order that they're drawn, they could just list them in ascending order every time. So is that how it's done? Do they list it in ascending order, or in draw order?

AFAIK most lotteries report their numbers in ascending order regardless of what order they were drawn in. It's more convenient to read them that way and it doesn't change the result wrt which ticket wins.

 

[alfaniner]

IMO this isn't a math question but a psychology question. The odds are random but human behavior is not necessarily random. Most people avoid sequences like that under the mistaken belief that it's somehow less likely than other combinations, however I don't find it surprising that there are at least some people attracted to it. Some may think "no one else would pick this" while others choose it because it's "easy" or "lucky".

Overall there are going to be combinations that people puck much more often than random chance would dictate and the winning numbers just hit one of them. 1, 2, 3, 4, 5 and Powerball 6 probabaly would have had even more winners.

I would be interested in seeing the breakdown of numbers chosen. (OK, it would be a big list -- maybe the top 100 that are often repeated.) I think 1-5 would be common, but more common would be Powerball 7 rather than 6. Because 7's "lucky".

 

[Ziggurat]

IMO this isn't a math question but a psychology question. The odds are random but human behavior is not necessarily random.

It's very difficult for people to be random even when asked to be.

 

[lomiller]

I would be interested in seeing the breakdown of numbers chosen. (OK, it would be a big list -- maybe the top 100 that are often repeated.) I think 1-5 would be common, but more common would be Powerball 7 rather than 6. Because 7's "lucky".

I think both would be more common than most combinations.


It's also worth noting that the winning combo this time is the numbers symmetrically around "lucky" 7. 5,6 ,7,8, 9 so it may be picked more frequently than 2,3,4,5,6.

Powerball being a separate selection is probabaly being tacked on at the beginning or end to get a 6 number sequence. Choosing 7 as the powerball selection could be even more common if it people were targeting the numbers around 7 for the first 5 numbers.

 

[TruthJonsen]

I would be interested in seeing the breakdown of numbers chosen. (OK, it would be a big list -- maybe the top 100 that are often repeated.) I think 1-5 would be common, but more common would be Powerball 7 rather than 6. Because 7's "lucky".

The link above had a link to statistics with some of that:

https://za.national-lottery.com/powerball/statistics

 

[Dr. Keith]

Indeed. There is no way to increase your chances of winning the lottery (apart from buying more tickets), but, if you must play, you can increase the chances of being the sole winner by picking a combination that no-one else has chosen. The frequency of chosen numbers is often available; as Lothian says, birthdays are very often used. Some lotteries even have the option for a random number selection which, I think, avoids combinations already picked.

I know this is not true locally and, if you think about it, it is probably not true anywhere. That would give a huge advantage. Well, "huge" in the very slim chances we are talking about where the odds are 1 in 25m.

Locally a certain number of entries can be printed on a single ticket. So, you may get a half dozen entries on one print out. Someone got the same entry twice on a single ticket. The lotto people said that it was not uncommon and the in the event of a multi-winner situation he would be awarded for both entries.

 

[zooterkin]

I know this is not true locally and, if you think about it, it is probably not true anywhere. That would give a huge advantage. Well, "huge" in the very slim chances we are talking about where the odds are 1 in 25m.

You're probably right, I'm not sure where I got the impression from.

I've only ever filled in a lottery ticket in the UK once, and it turned out to be the one day in the year you couldn't buy a ticket, so I've never actually taken part, and my knowledge is entirely theoretical.