Mystery Cash Problem - Best Time to Draw your Envelope?

Hi, I've been doing some searching and can't seem to find an answer to this problem, the closest I found was the "Draw the short straw" problem solution, but that stops the drawing when the short straw was drawn.

Here's my question, I'm hoping somebody can help answer it:

I sometimes receive a "Mystery Cash" reward where you arrive at the casino, get in line and everyone gets to draw one envelope out of a huge bin with a mystery amount in it. My simple question is: When is the best time to draw to get the highest possible prize/return?

Some more details: Usually there is one grand prize with huge number of the lowest prize, say $5, with a grand prize of $25,000. Let's use this distribution of prizes for this problem:
1000 total envelopes:
1 Grand Prize of $25,000
3 Prizes of $5,000
5 Prizes of $1,000
10 Prizes of $500
15 prizes of $100
25 prizes of $50
50 prizes of $25
891 remaining prizes of $5

Assuming that you don't know what prizes have already been pulled, and everyone gets a chance until the bin is empty, when is the best time to pull the envelope, in the beginning, in the middle of the promo or the end? Is there an optimal time to go or is it always the same odds to pull an envelope? The main factor is that you don't know, and nobody will tell you if somebody already won the $25,000 or not. Should I line up early, or let the line die out before going?

EDIT:

Thinking about it more, I guess the nagging point is there there has to be a "highest probability" for each of the 1000 draws of when the grand prize will be drawn since the probably goes up with each person NOT drawing the grand prize. The other factor is I'm thinking there has to be a "highest expected value" at some point. Maybe it's similar to the "Same birthday" problem when you get to 23 people gets to approx 50/50 chance.

Welcome to the forum! I'm sure there is a definitive mathematical answer, similar to scratch card lotto, where the drawing becomes +ev, but I think it would require knowing what has already happened.

Since they don't disclose that (and it seems odd they don't ), are you able, if you are there, to see or hear enough to know when the largest prizes are gone? Is the prize structure published in advance?

If you really have to go in blind, my OPINION is the best time, over several drawings, would be to be in line after 150 to 200 people have gone ahead of you, in your hypothetical pool of 1000.

Reason I find it odd is that, the places I play, they do the whole thing over the PA system and announce big winners as they are pulled. Seems like they give up some marketing advantage if they keep it secret.

Edit: I should say, since you're new, I'm not a math guy, but there are many on here, who can perhaps provide a better answer.

Yes, I think I'm wondering if there's a specific math answer that would determine it in the future, like the "Same Birthday" problem. I guess the point is I don't want to sit around and wait around the promo area if I don't have to. For instance, if the answer is usually the end is best, then I'll gamble for a while and show up at the end of the promo. If the first to draw is best, then I'll line up early.

I think ultimately, it was more of a theoretical exercise for me to understand if there's a way to maximize EV in this situation, or does it matter at all and every draw is the same odds.


EDIT:

The other piece of it is that since I don't have to pay for it, I will ALWAYS draw since I always make at least $5 for $0 investment (other than time). Just wondering if time is a factor.

It's not the same as the scratch ticket calculation because, at least in my state, you can find out what prizes have already been collected. The problem there is to determine the highest EV scratch ticket to buy based on that extra info. Obviously without any info buying scratch tickets is -EV.

This situation is completely different because you are getting a bonus prize which always has a positive EV. If I have counted correctly in your hypothetical example there is $57205 in prize money and 1000 tickets so the EV of a ticket is simply $57.205. If you knew which prizes had already been drawn (which you say you don't), you could calculate the EV of an undrawn ticket at that point, but that wouldn't help you choose when to draw. The big prize is equally likely to be drawn at any time. If you happen to know a lot of cheap prizes have been drawn then the EV of the remaining tickets is higher, but you couldn't have gotten that info without foregoing the chance to win the big prize should it have happened to be drawn in the first few tickets. When you draw doesn't affect your probability of winning. In the case where you don't know about the other tickets, it doesn't affect your experience at all because all you know is you drew a ticket and it won a certain amount. For this drawing I would just get my ticket when I didn't have to wait in line.

If the drawing was run such that I DID know the outcomes of previously drawn tickets, I would want to draw first. It wouldn't impact my probability of winning the big prize, but it would be less fun to play knowing the big prize has been drawn. Going first makes that scenario impossible.

Quote: kmj1104213

Assuming that you don't know what prizes have already been pulled, and everyone gets a chance until the bin is empty, when is the best time to pull the envelope, in the beginning, in the middle of the promo or the end?

It doesn't make any difference. Everyone has the same chances.

Even if you did know the prizes of everyone who picked before you, it doesn't help to jump in at a specific point.

Just take any envelope when the line is short and be done with it.

Quote: Wizard

It doesn't make any difference. Everyone has the same chances.

Even if you did know the prizes of everyone who picked before you, it doesn't help to jump in at a specific point.

Just take any envelope when the line is short and be done with it.

Huh? If there are 1000 entries and approx 900 of them are $5 and others are better... the drum of envelopes has memory then... if you see 500 $5's get drawn and NONE of the top prizes that nearly doubles your EV....

It all depends if you can know what others have drawn (i.e. know the remaining contents of the drum). If you can't, then it doesn't matter. If you really cared (probably a waste of time at this level) you could camp out next to the drum and just watch and see if anyone hits any of the big ones.

Quote: Romes

if you see 500 $5's get drawn and NONE of the top prizes that nearly doubles your EV....

If you see 500 entries drawn, it is equally likely that ALL of the top prizes are gone, which would cut the EV nearly in half.

Quote: Romes

Huh? If there are 1000 entries and approx 900 of them are $5 and others are better... the drum of envelopes has memory then... if you see 500 $5's get drawn and NONE of the top prizes that nearly doubles your EV....

It all depends if you can know what others have drawn (i.e. know the remaining contents of the drum). If you can't, then it doesn't matter. If you really cared (probably a waste of time at this level) you could camp out next to the drum and just watch and see if anyone hits any of the big ones.

That's true, but if you bide your time, all the big prizes might come out first and you would have a lower EV. I'm saying there is no strategy to this that increases EV on average. If you disagree, give me one.

It is like I've been saying about Deal or No Deal for years, if the banker gave fair offers, equal to the average of all unopened suitcases, there would be no way to game the game, not considering the fact that the utility of money is not linear.

Quote: Wizard

That's true, but if you bide your time, all the big prizes might come out first and you would have a lower EV. I'm saying there is no strategy to this that increases EV on average. If you disagree, give me one...

Hmm, I think I see your point. I was equating it to a blackjack shoe, and waiting until a better time to play... but I think the difference is with blackjack if the shoe doesn't get good, I'd just leave and go on to another table/shoe... with this, he's going to draw no matter what... i.e. play no matter what. Is that why it doesn't matter in the end?

My original thought was to equate it to a blackjack shoe, with memory of envelopes drawn and envelopes to remain.

I agree with Wizard. The timing of when you draw the envelope should make no difference.

If you wait a certain pre-determined amount of time before drawing and watch people open their envelopes you will see one of two things:

- someone screaming and celebrating because they got the big prize. You now know you waited too long.
- no one screaming and celebrating. You chances of getting the big prize have apparently increased.

The a priori strategy of waiting a certain time and then drawing will have the same average expectation/EV/outcome as drawing as early as possible or as late as possible or at random -because you must average over both of the above scenarios.

A potentially more interesting version of this problem would involve knowing the prior drawings and a cost of D dollars required in order to choose an envelope. This yields an additional option, where we don't choose an envelope at all (and save the D dollars). Then, there would be some optimal strategy of when to choose an envelope based on what has already been drawn.

As a trivial example, consider D = $20000, with the same envelopes as in the first post. Then, it would be best not to pick an envelope at all, except in the rare case where the $25000 envelope is left as the last one, guaranteeing a $5000 profit.

I haven't yet finished thinking about how to generate a strategy for an arbitrary choice of D, but I wanted to share in case anyone else found it interesting.

EDIT: After seeing Romes mention blackjack, I guess this scenario is akin to counting a shoe, and waiting to wong in for at most one hand?

counting in BJ is also worthless unless you can alter your betting. Nothing can be altered waiting for the envelope either.

So after reading all of your responses, I think you guys put me on the right track, and the Wizard is right (isn't he always?). I now agree that the timing is irrelevant, and the chances are the same, but only because you don't know the prior envelopes already drawn. Here's how I proved it to myself. I simplified the problem into 10 envelopes, 9 envelopes with $5 each and 1 envelope with $1000 (Grand Prize). This table shows that given that you know nothing about the other envelopes, the chance is the same at any given point.

Mostly self explanatory, but few notes, column E starts at 100% since it's impossible to have been drawn before the first round. This is the cumulative product of all probabilities that it was not found in every round before this. Column G represents the probability that you will draw this round TIMES the probability that it hasn't already been found.

Thanks everyone for your input and helping solve this problem.

(A) Round #	(B) Draw 1 Envelope	(C) Envelopes Remaining	(D) Probability of Drawing Grand Prize in this round, assuming not already drawn (B / C)	(E) Probability that the Envelope will NOT be drawn this round, assuming not already drawn (1 - D)	(F) Probability that Envelope has not been drawn before this round (Previous Value * E)	(G) Probability that you will draw grand prize this round (D * F)
1	1	10	10.0%	90.0%	100%	10%
2	1	9	11.1%	88.9%	90%	10%
3	1	8	12.5%	87.5%	80%	10%
4	1	7	14.3%	85.7%	70%	10%
5	1	6	16.7%	83.3%	60%	10%
6	1	5	20.0%	80.0%	50%	10%
7	1	4	25.0%	75.0%	40%	10%
8	1	3	33.3%	66.7%	30%	10%
9	1	2	50.0%	50.0%	20%	10%
10	1	1	100.0%	0.0%	10%	10%

I would assume that the big $25,000 envelope is on close to the bottom of the heap. Because this is just my educated theory, casinos want the big prize to be on the bottom, not the top to hold on to it as long as possible and want to build anticipation. If they put it on the top the anticipation fades really fast. There are 50 lucky players. Bobby and Sue are first. Bobby picks up the first one and it is a $500 envelope. He is happy. Sue picks up the second one. It is the lucky $25,000! She is ecstatic! She won the big $25,000! She says she is taking her family on a vacation after paying taxes and fees! There is thunderous applause from the Casino staff but the other 48 players grumble in bitter resentment , lamenting they will only win either $5 or $500 since the big win has been found.

Now imagine how much fun it will be if the $25,000 is the 49th envelope. The anticipation and excitement would be tremendous as each envelope is a $5 or $500 envelope. So many thoughts of,"I could be the $25,000 winner! Would be rampant. And when the 49th envelope is opened, the crowd goes wild and the winner is ecstatic!

Quote: Nathan

I would assume that the big $25,000 envelope is on close to the bottom of the heap. Because this is just my educated theory, casinos want the big prize to be on the bottom, not the top to hold on to it as long as possible and want to build anticipation. If they put it on the top the anticipation fades really fast. There are 50 lucky players. Bobby and Sue are first. Bobby picks up the first one and it is a $500 envelope. He is happy. Sue picks up the second one. It is the lucky $25,000! She is ecstatic! She won the big $25,000! She says she is taking her family on a vacation after paying taxes and fees! There is thunderous applause from the Casino staff but the other 48 players grumble in bitter resentment , lamenting they will only win either $5 or $500 since the big win has been found.

Now imagine how much fun it will be if the $25,000 is the 49th envelope. The anticipation and excitement would be tremendous as each envelope is a $5 or $500 envelope. So many thoughts of,"I could be the $25,000 winner! Would be rampant. And when the 49th envelope is opened, the crowd goes wild and the winner is ecstatic!

Your "educated theory"? Then you spin off a hypothetical situation that includes a likely illegal seeding of the random drawing process?

You don't get to present yourself to a new member as a gaming odds expert here, Nathan. Not your first time giving nonsense gaming advice to a new member, either. I can live with speculation or guessing; I do it myself, and I'm not a mathematician. But I don't reprepresent myself as an expert, either.

This site is about true odds and smart gaming. Keep it real.

Hey Babs! I’m going to back Nathan on this one. Id bet on the jp envelope not being on top as often as random would dictate. Likely illegal yes and I have little to no experience with this specific situation but it’s smart to question whether your getting a fair shake. Casinos do things illegally sometimes and they might not know it. Sometimes the players know the specifics of certain promos better than the employees. Can’t divulge too much but sometimes drawings (or other stuff) really are rigged without them knowing it because of faulty programming.

Quote: Mooseton

Hey Babs! I’m going to back Nathan on this one. Id bet on the jp envelope not being on top as often as random would dictate. Likely illegal yes and I have little to no experience with this specific situation but it’s smart to question whether your getting a fair shake. Casinos do things illegally sometimes and they might not know it. Sometimes the players know the specifics of certain promos better than the employees. Can’t divulge too much but sometimes drawings (or other stuff) really are rigged without them knowing it because of faulty programming.

I also somewhat agree with Nathan and what you said. She said it was a theory. It was a good theory(I found it to be a good post, especially for Nathan) and I think happens on occasion. Casinos oftentimes plan their drawings where the bigger prizes are drawn last, they do that for some of the reasons Nathan pointed out.

When there have been, you pick an envelope type promos that last days/weeks/ all month, I think there have been many times where the casinos wait to seed the drum with the bigger prizes near the end.


Your last sentence is very interesting.

Recently my wife and I participated in an electronic drawing(we were not playing for the drawing, we just happened to be earning a significant amount of entries) recently where I felt as if we were a victim of some kind of programming error that somehow excluded new card numbers from the drawing. I felt as if we should have been drawn during the first few drawings since the tickets could only be earned during that time and the casino was fairly dead at that point. It's really just a suspicion since the sample size is so small.

p.s. I really dislike electronic drawings. Mainly because they could exclude an AP's entries and they would never even know.

perhaps Nathan has been educated in the school of hard knocks

Quote: beachbumbabs

Your "educated theory"? Then you spin off a hypothetical situation that includes a likely illegal seeding of the random drawing process?

You don't get to present yourself to a new member as a gaming odds expert here, Nathan. Not your first time giving nonsense gaming advice to a new member, either. I can live with speculation or guessing; I do it myself, and I'm not a mathematician. But I don't reprepresent myself as an expert, either.

This site is about true odds and smart gaming. Keep it real.

It was an educated theory that Casinos would want to build excitement for that big $25,000 win by putting it near the bottom rather than near the top. ;)

It's been alleged in another thread that Ainsworth slots hold back mystery progressives until they are almost to the "must hit by" point. Let the record show that I said "alleged."

I did an electronic pull tab game for a small maker once. They would put two big prizes per batch but always hold back one until near the end.

Imagine that all the envelopes were handed out, then all opened at once.

Same thing. It doesn’t matter when.

Math puzzle time!

I know this was asked before, in other words, but this question makes me think of it.

You're in a casino promotion with 1,000 envelopes. The distribution of prizes in the envelopes is unknown, except no two have the same amount. Only the person who gets the top award gets to keep it, all others get nothing. It will be declared at the end what the top award is.

You are given the right of acceptance or refusal after every envelope opening, until you accept one. Once you accept an envelope, you're stuck with it.

What strategy would maximize the probability of getting the highest envelope? What would be the probability under that strategy?

Quote: Wizard

Math puzzle time!

I know this was asked before, in other words, but this question makes me think of it.

You're in a casino promotion with 1,000 envelopes. The distribution of prizes in the envelopes is unknown, except no two have the same amount. Only the person who gets the top award gets to keep it, all others get nothing. It will be declared at the end what the top award is.

You are given the right of acceptance or refusal after every envelope opening, until you accept one. Once you accept an envelope, you're stuck with it.

What strategy would maximize the probability of getting the highest envelope? What would be the probability under that strategy?

Since we know the actual distribution of envelopes in the posed problem, I'm still interested in the variation I posed earlier, where we have to pay some cost D for selecting an envelope. It also allows us to try to maximize EV rather than the probability of finding the best envelope.

EDIT: Attempted to add spoiler tags.

Best time to draw is obviously when it requires the least amount of time in line, unless you're trying to kill time.

Quote: Wizard

It's been alleged in another thread that Ainsworth slots hold back mystery progressives until they are almost to the "must hit by" point. Let the record show that I said "alleged." ..

I'll go ahead and state that it's true... I really can't fathom how no one has sued them because I've read in their machine rules and it does say the mystery award is given out "randomly" (which it most clearly is not).

Quote: Romes

I'll go ahead and state that it's true... I really can't fathom how no one has sued them because I've read in their machine rules and it does say the mystery award is given out "randomly" (which it most clearly is not).

I agree with you, but I don't think you can point to a specific law they are breaking. It is my understanding these gaffed machines still have a very small chance of hitting the Mystery Jackpot when it is low.

If I had anything to do with it, the help files should make it clear how the jackpot trigger works. Just saying it is "random" is not enough.

Quote: djtehch34t

Show Spoiler

This is the secretary problem. Pass on the first 1000/e ~= 368 envelopes. Then pick the first one that is bigger than the best from the first batch. The probability is 1/e ~= .368 of choosing the best envelope, though I'd have to work a bit to reconstruct the proof.

You're absolutely right!

In the past, you had to pick a card from a hat, as bad as I wanted to win the top prize, $100 was face up visible and I had to take it knowing there were many lower.

Quote: Romes

Quote: Wizard

It's been alleged in another thread that Ainsworth slots hold back mystery progressives until they are almost to the "must hit by" point. Let the record show that I said "alleged." ..

I'll go ahead and state that it's true... I really can't fathom how no one has sued them because I've read in their machine rules and it does say the mystery award is given out "randomly" (which it most clearly is not).

Random doesn't have to be fair, random doesn't mean fair. The phrase completely random implies fairness, but just random does not.