Statistics Question - Lottery Odds

[Gunpowder]

Ok so I'm trying to figure out the lottery odds for my fantasy football draft and can't quite seem to get it. We have 12 teams, last year's #1 gets 1 ball in the machine up to the last place team getting 12, so 78 total. In trying to figure out the odds I've come up with the following:

image.png (68.5 KiB) Viewed 486 times

The top is the average odds for each team to get each pick, and it appears that I'm somewhat on perhaps the right track somewhere but the totals are obviously off. They don't total to 100% and I have some negatives.

The bottom table is an attempt to find the expected remaining balls in each round, as obviously every time a team is selected there will be a corresponding reduction. I'm assuming that this is where I'm wrong - each round is totaled by multiplying the average odds of each team to have been selected and the amount of lottery balls that would on average remove from the pool.

I've never studied any sort of statistics so this is well over my head. A quick internet search suggests something called a Plackett-Luce model, which looks like a lot of difficulty to learn and for something quite trivial. Does anybody have any experience/knowledge of things like this?

[Shirley]

The odds of getting a pick in round two equal the odds of the number of balls divided by 77 (78-1) times the odds of NOT getting the pick in round 1. So for the 12-ball guy, it’s (66/78)*(12*77). For round 3, it’s the round 3 odds (76 balls left) times the NOT odds for rounds 1 & 2. Repeat that pattern.

[Gunpowder]

The odds of getting a pick in round two equal the odds of the number of balls divided by 77 (78-1) times the odds of NOT getting the pick in round 1. So for the 12-ball guy, it’s (66/78)*(12*77). For round 3, it’s the round 3 odds (76 balls left) times the NOT odds for rounds 1 & 2. Repeat that pattern.

I figured trying to account for the other teams was the error. I'll give that a shot.

EDIT: Ahhh, right, should have just removed one ball per round and ignored a pick for a team that already went (since we just draw again). Thanks Shirls

[Gunpowder]

Still struggling with this - what are for example the 1-ball guy's (lolz) odds through all 12 rounds? I have him at 1.28% for every round, which obvs doesn't add up to 1. The other player's odds do change from round to round but still don't equal 1.

Am I not accounting for other teams being removed from the pool?

[mister d]

I think I tried to do this once and gave up because each round compounded on itself. Its not just your odds less your not odds because, like you said, that would keep them flat throughout. Its something like your odds of not having been picked in the prior round times every other individual odds (because the denominator changes depending on who is out) and then do that over and over. It went beyond what I cared to do in excel for fantasy football.

[Gunpowder]

I think I tried to do this once and gave up because each round compounded on itself. Its not just your odds less your not odds because, like you said, that would keep them flat throughout. Its something like your odds of not having been picked in the prior round times every other individual odds (because the denominator changes depending on who is out) and then do that over and over. It went beyond what I cared to do in excel for fantasy football.

Yep - I thought I was really close the first time through but I was doing something wrong. I was doing the top portion correct (per Shirley's post) but wasn't considering the one ball taken per round.

I'm gonna fuckin' make this shit work though. IF IT TAKES ME ALL GD YEAR

[Gunpowder]

The Plackett-Luce model btw has sigma functions so I think you're right in how it needs to be done.

[A_B]

Wouldn't there have to be a new set of odds created every time a ball is picked? Then you'd have to account for every order in which they could be picked as well? And if you aren't removing each person's balls once they are picked, you're going to get a bunch of fucked up calculations that would never really be right? Or I guess this last just considers them to be phantoms and not really in the drawing even though each of those balls can still get picked it throws off the overall odds but not the relative odds.

I don't know why are you asking me for?

[Gunpowder]

Wouldn't there have to be a new set of odds created every time a ball is picked? Then you'd have to account for every order in which they could be picked as well? And if you aren't removing each person's balls once they are picked, you're going to get a bunch of fucked up calculations that would never really be right? Or I guess this last just considers them to be phantoms and not really in the drawing even though each of those balls can still get picked it throws off the overall odds but not the relative odds.

I don't know why are you asking me for?

Pre-draft, it would be a set number depending on other teams' probabilities. ie before a single ball is drawn, each team's odds for each pick are going to be static. Once you start picking you have to account for the prior selections but that's something I would do live and that's very easy to calculate.

[mister d]

Right. Live you can do this as a single set of calcs after each pick. Odds right now of the #1 odds team getting the 7th pick are "fuck this" difficult.

[Shirley]

The complicating thing for rounds beyond round 1 are that you don't know how many balls are being removed in each round. When I typed that last night, I was thinking one ball removed each round, but that's not right. The number removed is unknown, because it depends on who got picked first. So, to do this right, you'd have to set up a pretty complex model, accounting for each team's chances in the prior round - a fucking nightmare.

Round 1 = easy
Round 2 = hard
Round 3 = super hard ...

It would almost be easier to write a short program that understands how to remove the balls after picks and then Monte Carlo that thing to calculate the true odds.

[Gunpowder]

That's what I attempted to do in my bottom table but didn't quite get it. It multiplies each teams odds that round * the balls removed (so for team 12 15% * 12, etc) for each round. I actually get the odds for each round to add up to 1 but they just a little bit off overall.

I don't think I have the capacity to write a Monte Carlo simulator...too bad I don't remember any of that cutting edge Fortran.

[Gunpowder]

I know it's needlessly complicated but so were my stock matrixes and I am just particular about my sexy Excel files.

[Shirley]

That's what I attempted to do in my bottom table but didn't quite get it. It multiplies each teams odds that round * the balls removed (so for team 12 15% * 12, etc) for each round. I actually get the odds for each round to add up to 1 but they just a little bit off overall.

I don't think I have the capacity to write a Monte Carlo simulator...too bad I don't remember any of that cutting edge Fortran.

Good time to learn a little Python!

[mister d]

So this is the manual way to do Rd 2, right? And then 3rd pick would have to account for every possible combination of two teams removed, 4th pick would be every combination of three teams removed, etc, etc ...

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(Last column is weighted average based on the 1st round odds)

[degenerasian]

This seems impossible to me because you don't know how many balls you are removing each round

[Giff]

Is this a place for probably a really dumb math question? Let's say something happens 25 out of 100 times. If there is a 438% increase in that thing happening, how many times out of a 100 would it happen?

[mister d]

This seems impossible to me because you don't know how many balls you are removing each round

Right, that's why you have to account for every possibility every round. I've now done 3 rounds and am getting close to continuing "for fun".

[degenerasian]

This seems impossible to me because you don't know how many balls you are removing each round

Right, that's why you have to account for every possibility every round. I've now done 3 rounds and am getting close to continuing "for fun".

but you're doing it manually? or a bunch of case statements?

[A_B]

Is this a place for probably a really dumb math question? Let's say something happens 25 out of 100 times. If there is a 438% increase in that thing happening, how many times out of a 100 would it happen?

109.5 times.

Or wait...maybe 126.095 times

That's not right either.

134.5 Final answer.

[Shirley]

Is this a place for probably a really dumb math question? Let's say something happens 25 out of 100 times. If there is a 438% increase in that thing happening, how many times out of a 100 would it happen?

25(5.38) = 134.5 times out of 100. Which of course, isn't really possible.

[mister d]

Is this a place for probably a really dumb math question? Let's say something happens 25 out of 100 times. If there is a 438% increase in that thing happening, how many times out of a 100 would it happen?

Doesn't it max out at a 300% increase? 50 out of 100 is a 100% increase, 75 is 200%, 100 is 300%.

[A_B]

Is this a place for probably a really dumb math question? Let's say something happens 25 out of 100 times. If there is a 438% increase in that thing happening, how many times out of a 100 would it happen?

Doesn't it max out at a 300% increase? 50 out of 100 is a 100% increase, 75 is 200%, 100 is 300%.

Maybe he's working on margin!

[Giff]

I'm working on a response to an idiot who posted vaccines have caused a 438% increase in miscarriages using a stat that says 15-25% of all reported pregnancies end with a miscarry. That number is absolutely ridiculous and all fucking vaccines would have stopped.

[mister d]

Just ask them to cite the pre and post percentages. You'll quickly see if they're maybe mistating a real stat (438% increase of miscarrying in day 5 post-vaccine or something) or just a dumb piece of shit.

[Giff]

Just ask them to cite the pre and post percentages. You'll quickly see if they're maybe mistating a real stat (438% increase of miscarrying in day 5 post-vaccine or something) or just a dumb piece of shit.

Oh I know the answer. And rethinking going back anyway because I've calmed down. He never even posted a source, despite his assertions it was just as reputable as the one I posted. Never mind a doctor posting a 1000 word response below me about how there is no current evidence showing an increase in miscarriages caused by vaccines...

[Gunpowder]

This seems impossible to me because you don't know how many balls you are removing each round

But you do (theoretically) know the odds of removing each amount which should be enough (I'm assuming) to calculate expected odds. Could be wrong.

[Gunpowder]

I figured out a way to get damn close to it and with a little manual fudging it will pass as proper odds. If someone at the fantasy football draft decides to check my work with their own analysis, more power to them and I'd offer my support.

Anyway, what I was doing to approximate the summing functions was calculating the "expected" balls removed for each team by round, ie the total balls remaining after round 5 would be the odds of team 12 having been selected times their 12 balls (in this case a calculated 65% times 12 or 7.86 balls), team 11's chances * 11 balls, and so on. This gave me 38.54 expected balls remaining after the 5th round. HOWEVAH, the odds only apply to the other teams. Ie if we're calculating team 12's odds in round 6, I use the expected balls for the other teams, BUT we know that all 12 of team 12's hot juicy balls are remaining (as opposed to 7.86 for the other teams). So for each team I removed their odds and that seemed to be generally on the right track. Where I think I went wrong is what I did next - you can't just remove team 12's balls in a vacuum, you have to adjust the other teams accordingly. I summed all of the expected percentages for the remaining teams and averaged the value removed from team 12 over the remaining 11 teams. Clearly this wouldn't be this clean since team 1 has different odds than team 5 and you probably have to account for one team being removed each round, but it at least seems to finally follow the proper patterns.

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This is where I'm going to leave it after I make a few slight adjustments to get all the teams to 100%.

[mister d]

Ooo ... I'm going to do a quick Monte Carlo and lets see how close we are. Everyone stay turned!

[mister d]

1,250x sim monte carlo:

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[Gunpowder]

Wait wtf you can do Monte Carlo sims in excel???? I would have just done that.

That looks pretty close to the final version I had after I manually changed it.

[Shirley]

51% chance of getting the last pick for the one-ball guy doesn't sound right to me. How did you set up the Monte Carlo?

[mister d]

Yeah, I thought that was a bit weird too but the odds are pretty stacked against a 1 ball in 78 to start team? Even head to heads against just teams 2 and 3, they're a 1/6 shot.

I built it out as a 12 column, 78 ()RAND matrix where the first column would find the MAX number, that would be the "winner" and each column to the left would eliminate those teams already picked. So if it went 12 1st and 7 2nd, the 3rd column would blank out the 19 rows with those as possible winners and find the MAX RAND of the remaining 59 options. (And then I copied that setup 1,249 times across the columns until I ran out and did SUMIFS to record each result.)

[mister d]

Maybe it does make sense? If you just look at it from the 1 ball perspective and assume the highest odds keep winning, they aren't even a 1/10 shot until the 9th pick. I don't know.

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[Gunpowder]

Yeah, I thought that was a bit weird too but the odds are pretty stacked against a 1 ball in 78 to start team? Even head to heads against just teams 2 and 3, they're a 1/6 shot.

I built it out as a 12 column, 78 ()RAND matrix where the first column would find the MAX number, that would be the "winner" and each column to the left would eliminate those teams already picked. So if it went 12 1st and 7 2nd, the 3rd column would blank out the 19 rows with those as possible winners and find the MAX RAND of the remaining 59 options. (And then I copied that setup 1,249 times across the columns until I ran out and did SUMIFS to record each result.)

I like it.

There's no way to use a random number that doesn't update every single task, right?

[mister d]

Don't think so. When I want to freeze results I just do a second tab that I can hardcode.

[Gunpowder]

Don't think so. When I want to freeze results I just do a second tab that I can hardcode.

That's how I did this one, with a macro button to make it seem like it wasn't rigged together with figurative duct tape.

[brian]

Powerball and Mega Millions over a half bill so spending the night in Laughlin and gonna head over the river to AZ to buy some tickets. (Have a $10 winner from April to cash in so mostly playing with house money anyway)

[brian]

Said winner was actually for $200 it turns out ( I had 3 & the mega ball, thought I only had 3 for $10 so turns out definitely free rolling.

[Steve of phpBB]

The Cubs beat the Yankees in New York tonight.

For the first time ever.