When Should I Draw?

[CCass]

Assume that you are one of 12 participants in a blind draw. Each of the 12 participants paid $100, and there is 1 winning ticket worth $1,200. Mathematically, what is the best position to draw from? Is there any advantage to any of the positions?

Now assume that there are 12 tickets and only one winner, but you can see the other draws as they happen, and if the "winner" is drawn, you don't have to "buy" your chance, and can therefore save your $100. What is the best position to draw from in this scenario?

I have attempted to compute answers to both of these scenarios, but I fear my math skills are lacking, so I have turned to the experts to either concur with my findings, or expose my shortcomings.

[dtbog]

Well, I'm no expert, but...

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Mathematically, what is the best position to draw from? Is there any advantage to any of the positions?

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Of course not! Everyone has a 1/12 chance of winning.

If you think of it this way: if you're 5th to draw,

( 4/12 = 1/3 chance that the winner has already been drawn ) * (0 chance of you winning, given this scenario) = 0% chance of winning in this situation

( 8/12 = 2/3 chance that the winner has not already been drawn ) * (1/8 chance of you winning, given this scenario) = 2/24 = 1/12 chance of winning.

So, total sum = 1/12, same as you'd expect.

... and so on for the rest of the positions. Makes no difference.

[ QUOTE ]
Now assume that there are 12 tickets and only one winner, but you can see the other draws as they happen, and if the "winner" is drawn, you don't have to "buy" your chance, and can therefore save your $100. What is the best position to draw from in this scenario?

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This scenario doesn't really make any sense in terms of EV calculation, because if the first person to draw selects the winner, where does the prize come from? No one else bought tickets.

-DB

[CCass]

When I did the math for scenario #1, I got the same answer as you, it makes no difference when you draw.

As for scenario #2, I somewhat agree with your answer, so I should explain a little more fully. We are discussing running a pool here at work where 12 people get 1 random draw out of a hat filled with the names of the 12 NFL playoff teams. Assuming I think a particular team is a clear favorite to win, I would have no incentive to draw if they were already taken. That is why I have proposed that all 12 people put up the money before we pick (scenario #1) and make it truely random. But the other participants may not go for this idea, and I was trying to figure out if there was a more preferred position to pick from if I had the option to "drop out" if my team was gone before I chose.

[dtbog]

EV of the second scenario, assuming that the prize pool is based on the number of previous ticket-buyers and given that you will get to draw:

drawing first: (-100)*(11/12) + (0)*(1/12) = -$91.67
second: (-100)*(10/11) + (+100)*(1/11) = -$81.81
third: (-100)*(9/10) + (+200)*(1/10) = -$70.00
fourth: (-100)*(8/9) + (+300)*(1/9) = -$55.55
fifth: (-100)*(7/8) + (+400)*(1/8) = -$37.50
sixth: (-100)*(6/7) + (+500)*(1/7) = -$14.28
seventh: (-100)*(5/6) + (+600)*(1/6) = +$16.67
eighth: (-100)*(4/5) + (+700)*(1/5) = +$60.00
ninth: (-100)*(3/4) + (+800)*(1/4) = +$125.00
tenth: (-100)*(2/3) + (+900)*(1/3) = +$233.33
eleventh-100)*(1/2) + (+1000)*(1/2) = +$450.00
twelfth: obviously, +$1100

[dtbog]

Wait a second, here...

...so you want to run an office pool where the teams are to be chosen randomly, but you want to have the option of dropping out unless you get the team you want?

Imagine if everyone felt that way. There would clearly be no office pool at all.

In the first scenario you're describing, it's literally just a random gamble for $100 to win $1200, but it gives you some excitement during the football playoffs. This could be fun.

In the second scenario, though, you're basically looking for a way to only enter the raffle if you have a better chance of winning than everyone else. Seems a little edgy to me? =)

Personally, I wouldn't enter a raffle like that if everyone hadn't already paid. What if you got together a group of 12, started picking names, and then two schmucks snuck out the back of the room when they saw the Eagles get chosen. Now you have 10 people in for $100 each, and two un-chosen teams.

...how do you assign people to these teams? Walk around the office saying "who wants a $100 raffle ticket for the 8-8 Rams?"

I don't see how it could work this way. At all.
-DB

[CCass]

[ QUOTE ]
Wait a second, here...

...so you want to run an office pool where the teams are to be chosen randomly, but you want to have the option of dropping out unless you get the team you want?

Imagine if everyone felt that way. There would clearly be no office pool at all.

In the first scenario you're describing, it's literally just a random gamble for $100 to win $1200, but it gives you some excitement during the football playoffs. This could be fun.

In the second scenario, though, you're basically looking for a way to only enter the raffle if you have a better chance of winning than everyone else. Seems a little edgy to me? =)

Personally, I wouldn't enter a raffle like that if everyone hadn't already paid. What if you got together a group of 12, started picking names, and then two schmucks snuck out the back of the room when they saw the Eagles get chosen. Now you have 10 people in for $100 each, and two un-chosen teams.

...how do you assign people to these teams? Walk around the office saying "who wants a $100 raffle ticket for the 8-8 Rams?"

I don't see how it could work this way. At all.
-DB

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Essentially, you and I agree. In the first scenario it is simply luck of the draw, which doesn't really interest me. Much like playing poker, I want an edge.

I also agree that the second scenario could be problematic because there will be teams that no one wants, and if the good teams go early, there goes the pool. I have already told the other guys that if we are going to do the pool, we need to get the money up front, then have everybody draw at the same time (scenario #1). I offered scenario #2 just to see if there would be a mathematical advantage to drawing in a certain order if you knew what had already been drawn.

[gaming_mouse]

[ QUOTE ]
. I have already told the other guys that if we are going to do the pool, we need to get the money up front, then have everybody draw at the same time (scenario #1).

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Yes, obviously you need to put the money in up front. Once this is done, though, it makes no difference whether or not you draw at the same time. Once everybody's money is locked up, you can draw in turn from a hat (mixing it up each time). You can even show each other what you've picked. Of course, if the first person drawing picks the best team, then already he has the best chance of winning.

But before any draws have been made the system I described gives everyone an equal chance of winning. And that is all you are trying to do.

gm