Request: Probability Calculation

[Dirk Diggler]

Doing a sweep at work, choose 6 numbers for Lotto. Your numbers get recorded/highlighted on a spreadsheet each draw and the first person to get all their numbers highlighted will scoop the prize, which is everyone's entry fee.

£20 to enter
53 people have participated
Jackpot is £1060

2 participants, who are on opposite shifts, have chosen the same 6 numbers.

Edit to add: this isn't theoretical, it actually happened this week.

 

[bloodiedathame]

how many numbers can they choose from?

 

[Dirk Diggler]

how many numbers can they choose from?

It's UK lotto they've chosen from, so 59.

 

[bloodiedathame]

Roughly 1 in 32,700 chance (not my maths).

 

[Zefan]

So the only information that is relevant is the amount of numbers you pick, the range of numbers, and the amount of unique selections. The exact odds will be impacted if depending on whether the full range is evenly selected, for example if half of the entries picked 7 and only 1 person picked 59, you can see how this will skew things. Realistically the odds are so vanishingly small even in the best of conditions so I wouldn't worry about it.

As you are just playing the exact same game as lotto, you can just use odds of winning lotto (1 in 45 million), and then divide it by the amount of entries. This works out to my maths as being 1 in 866k. This is with a completely even spread of selections.

Just rereading OP and I think I missed something, does the highlight remain from previous draws? If so this is wildly different and I don't have the brain time to work it out right now

 

[wonko]

Are you looking for the chances of someone picking a set of numbers that have already been picked? Pre-coffee brain gets the same as Zefan:
Combinations of randomly picking 6 from 59 = 45057474. Probability of one person picking the same set as 52 others = 1 in (45057474 / 52) = 1 in 866K

If you're looking for the chances of any 2 sets of numbers from 53 being the same, there are 1378 pairs which reduces the odds significantly - a bit like the birthday paradox where you only need 23 people for there to be a 50% chance that two share the same birthday.

Or somethings like that... This assumes that people pick numbers randomly, which they don't.

 

[Mr Jack]

People don't actually pick numbers randomly. That makes this much more likely than a crude statistical calculation implies.

 

[Dirk Diggler]

Are you looking for the chances of someone picking a set of numbers that have already been picked?

Yes, essentially what's the chances of someone picking the same numbers as someone else.

Logic tells me that it's got to be the same odds of someone winning the lottery, because person "A" is representative of any random pick and person "B" is representative of the Lotto machine's pick. The fact that there's only 53 people in this pool is irrelevant.

 

[Mr Jack]

Logic tells me that it's got to be the same odds of someone winning the lottery, because person "A" is representative of any random pick and person "B" is representative of the Lotto machine's pick. The fact that there's only 53 people in this pool is irrelevant.

This is incorrect, because you have to consider every combination of two people within the pool. Have a look at the Birthday problem for a similar example of where your intuition is wrong.

 

[bloodiedathame]

I absolutely hate probability. I was a maths mechanics guy. Reading up on it a bit seems to suggest:

Calculation goes:
1. Total Number of Possible Sets: 45mil

2. Probability of No Two People Picking the Same Set:
The first person can pick any of the 45,057,474 sets, the second person must pick a different set, so they have 45,057,473 choices, and so on, up to the 53rd person who has 45,057,422 choices

P(all different)=45057473/45057474 × 45057472/45057474 etc

99.99694%

3. Probability of At Least Two People Picking the Same Set:
The complement of the probability that all pick different sets:
So, P(at least two same) = 1 - P(all different)

0.00306% (or 1 in 30,600).

 

[wonko]

Yes, essentially what's the chances of someone picking the same numbers as someone else.

Ah... do you mean picking the same numbers as a specific person, or any one of the other 52 people?
If it's the latter, then there are (almost) 52 times the chances of a match.

 

[TommyV]

I’ve read this whole thread, and now my head is hurting

 

[Haggisman]

People don't actually pick numbers randomly. That makes this much more likely than a crude statistical calculation implies.

Yup, you'll find that numbers 1-12 are significantly more frequent than anything else, 13-30 a bit less so, but still noticeably more frequent, and 31 a bit less again (not sure if there will be a noticeable difference between 29-30?)

 

[Psiko]

I absolutely hate probability. I was a maths mechanics guy. Reading up on it a bit seems to suggest:

Calculation goes:
1. Total Number of Possible Sets: 45mil

2. Probability of No Two People Picking the Same Set:
The first person can pick any of the 45,057,474 sets, the second person must pick a different set, so they have 45,057,473 choices, and so on, up to the 53rd person who has 45,057,422 choices

P(all different)=45057473/45057474 × 45057472/45057474 etc

99.99694%

3. Probability of At Least Two People Picking the Same Set:
The complement of the probability that all pick different sets:
So, P(at least two same) = 1 - P(all different)

0.00306% (or 1 in 30,600).

This is the correct answer if people pick their numbers completely at random. If you add in some bias for picking months, dates, etc then maybe double or triple the probability, at an guess? So 1 in 10000 chance if you assume a high bias. Still a very small chance so quite amazing that it happened!

 

[Mr Jack]

So 1 in 10000 chance if you assume a high bias. Still a very small chance so quite amazing that it happened!

Depends what you consider the probability over. Amazing that it happened to Dirk's club on this occasion, yup, amazing that it happened to a club like this over the many times they pick numbers? Not really. Or amazing that such an unlikely event happened to Dirk during his life (or even this year) not really.

Large numbers make unlikely things likely. Just as it's amazing that John McWinner won the lottery but unsurprising that someone did.