Monty Hall, we have a PROBLEM

[owslachief]

Re: Monty Hall, we have a PROBLEM

You've just ordered 13 bottles of the most expensive wine (half your paycheck went to it) for a daughter's wedding reception. As you near the big day, the Italian company you bought them places a frantic phone call to you telling you they realized that ONE of the bottles they sent is tainted. It came from their lab, which was experimenting with ways to neutralize the effects of certain biological hazards ... and somehow it got mixed in with your order. He further tells you that once ingested, the poison won't take effect until 24 hours later. One problem is there is NO WAY to distinguish the poisoned bottle from the other 12. It looks, smells, feels the same as the untainted wine. The company offers to fully replace your order but you tell them "my daughter's wedding is less than two days away! It'll never arrive on time!". Panicked, you then experience the calming affects of a sudden idea: A friend of yours owns a laboratory which just happens to have four live rats. With about 36 hours left to the reception, you're brimming with confidence your wedding guests can enjoy your favorite wine after all, and live to tell about it.

How does your friend at the lab proceed?

 

[bigblue_dl]

Re: Monty Hall, we have a PROBLEM

You've just ordered 13 bottles of the most expensive wine (half your paycheck went to it) for a daughter's wedding reception. As you near the big day, the Italian company you bought them places a frantic phone call to you telling you they realized that ONE of the bottles they sent is tainted. It came from their lab, which was experimenting with ways to neutralize the effects of certain biological hazards ... and somehow it got mixed in with your order. He further tells you that once ingested, the poison won't take effect until 24 hours later. One problem is there is NO WAY to distinguish the poisoned bottle from the other 12. It looks, smells, feels the same as the untainted wine. The company offers to fully replace your order but you tell them "my daughter's wedding is less than two days away! It'll never arrive on time!". Panicked, you then experience the calming affects of a sudden idea: A friend of yours owns a laboratory which just happens to have four live rats. With about 36 hours left to the reception, you're brimming with confidence your wedding guests can enjoy your favorite wine after all, and live to tell about it.

How does your friend at the lab proceed?

Dumps out all of the expensive wine, buys 5 boxes of Franzia, and refills them. No one knows the difference.

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

You've just ordered 13 bottles of the most expensive wine (half your paycheck went to it) for a daughter's wedding reception. As you near the big day, the Italian company you bought them places a frantic phone call to you telling you they realized that ONE of the bottles they sent is tainted. It came from their lab, which was experimenting with ways to neutralize the effects of certain biological hazards ... and somehow it got mixed in with your order. He further tells you that once ingested, the poison won't take effect until 24 hours later. One problem is there is NO WAY to distinguish the poisoned bottle from the other 12. It looks, smells, feels the same as the untainted wine. The company offers to fully replace your order but you tell them "my daughter's wedding is less than two days away! It'll never arrive on time!". Panicked, you then experience the calming affects of a sudden idea: A friend of yours owns a laboratory which just happens to have four live rats. With about 36 hours left to the reception, you're brimming with confidence your wedding guests can enjoy your favorite wine after all, and live to tell about it.

How does your friend at the lab proceed?

Assuming that there is no problem associated with opening a bottle and then re-corking it right away, then

Answer in white:

Open four bottles and serve each of the rats enough wine from one bottle to be lethal in 24 hours. 5-1/2 hours later or so, open four more bottles and serve each of the rats wine from one of those bottles. another 5-1/2 hours later or so, open four more bottles and serve each of the rats wine from one of those bottles.

If one of the rats is dead 24 hours later, the poison was in the first batch of four. If one of the rats is dead 29-1/2 hours later, the poison was in the 2nd batch of four. If one of rats is dead 35 hours later, the poison was in the 3rd batch. If they are all alive at the start of the reception, the poison is in the unopened bottle.

 

[aparch]

You've just ordered 13 bottles of the most expensive wine (half your paycheck went to it) for a daughter's wedding reception. As you near the big day, the Italian company you bought them places a frantic phone call to you telling you they realized that ONE of the bottles they sent is tainted. It came from their lab, which was experimenting with ways to neutralize the effects of certain biological hazards ... and somehow it got mixed in with your order. He further tells you that once ingested, the poison won't take effect until 24 hours later. One problem is there is NO WAY to distinguish the poisoned bottle from the other 12. It looks, smells, feels the same as the untainted wine. The company offers to fully replace your order but you tell them "my daughter's wedding is less than two days away! It'll never arrive on time!". Panicked, you then experience the calming affects of a sudden idea: A friend of yours owns a laboratory which just happens to have four live rats. With about 36 hours left to the reception, you're brimming with confidence your wedding guests can enjoy your favorite wine after all, and live to tell about it.

How does your friend at the lab proceed?

My guess: Split up the case into groups of 3, 3, 3, 2, and 2 bottles. Assign each rat to a group, and since I'm a gambling man, I leave one set of three bottles without a rat.

Give each rat samples from their bottles, and await results. Worst case; you have four hung over rats and three un-sampled bottles you throw away. Otherwise, throw out the two (or three) bottles from whichever rat is dead.


Is there an option to run out any buy your buddy eight more rats?

 

[unofan]

You've just ordered 13 bottles of the most expensive wine (half your paycheck went to it) for a daughter's wedding reception. As you near the big day, the Italian company you bought them places a frantic phone call to you telling you they realized that ONE of the bottles they sent is tainted. It came from their lab, which was experimenting with ways to neutralize the effects of certain biological hazards ... and somehow it got mixed in with your order. He further tells you that once ingested, the poison won't take effect until 24 hours later. One problem is there is NO WAY to distinguish the poisoned bottle from the other 12. It looks, smells, feels the same as the untainted wine. The company offers to fully replace your order but you tell them "my daughter's wedding is less than two days away! It'll never arrive on time!". Panicked, you then experience the calming affects of a sudden idea: A friend of yours owns a laboratory which just happens to have four live rats. With about 36 hours left to the reception, you're brimming with confidence your wedding guests can enjoy your favorite wine after all, and live to tell about it.

How does your friend at the lab proceed?

short and dirty version from the phone: You take the 13 samples and assign them a binary number from 0000 on up. You then give them to the rats in accordance with the associated binary number. So whatever one is 0101 you'd give to the first and third rat, for example. Based on which rats die and which live, you can tell which wine is bad.

 

[Twitch Boy]

Re: Monty Hall, we have a PROBLEM

Incidentally, blowing these problems wide open at teambuilding/other seminars is one of my favorite little joys in life.

ACTIVITY
Let's Make A Deal Problem (15 minutes)

Corpie hippy dippy presenter: "You're on Let's Make A Deal..."
Me: "Yes you should switch because it's 2 in 3 you didn't pick the car and Monty's locked into revealing a goat no matter what."
CHDP: "...fark."
Me: "Early lunch, team?"

 

[joecct]

Re: Monty Hall, we have a PROBLEM

You've just ordered 13 bottles of the most expensive wine (half your paycheck went to it) for a daughter's wedding reception. As you near the big day, the Italian company you bought them places a frantic phone call to you telling you they realized that ONE of the bottles they sent is tainted. It came from their lab, which was experimenting with ways to neutralize the effects of certain biological hazards ... and somehow it got mixed in with your order. He further tells you that once ingested, the poison won't take effect until 24 hours later. One problem is there is NO WAY to distinguish the poisoned bottle from the other 12. It looks, smells, feels the same as the untainted wine. The company offers to fully replace your order but you tell them "my daughter's wedding is less than two days away! It'll never arrive on time!". Panicked, you then experience the calming affects of a sudden idea: A friend of yours owns a laboratory which just happens to have four live rats. With about 36 hours left to the reception, you're brimming with confidence your wedding guests can enjoy your favorite wine after all, and live to tell about it.

How does your friend at the lab proceed?

Is the answer Jesus?

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

Here is a question that I am curious about and I don't quite know how to solve, though I feel close.

You read the news stories about lottery winners, and think it might be fun to get in on the action.

There are five numbers drawn, from 1 to 59 inclusive, and then a sixth number, from 1 to 35 inclusive. You want to match as many numbers as you can, especially the sixth one.

The question: which gives you better odds, if either one does:
a) do you pick the same five numbers / one number each time?
b) do you allow the game computer to generate random numbers for you each time?


My intuition (which is always suspect in these situations, eh?) is that, for one week, it won't matter. But what about over the course of a year, if you play twice weekly? Would the first give slightly better odds since you are trying to match a single set of set of specific numbers to a random draw, while in the second option both your guess and the winners are changing each time?

To state it differently, I know how to calculate the odds for the first option, but I'm not sure if that calculation methodology would be valid for the second option.


Let's extend it a bit further, assuming you are going to buy, say, five tickets for each draw:
a) do you pick the same five numbers for the first set five times, and change only the sixth number for your five draws?
b) do you allow the game computer to generate randome numbers for you for each ticket?


If you match only the sixth number you get twice what you paid for the ticket. Match all six and you win millions or maybe tens of millions.

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

Is there an option to run out any buy your buddy eight more rats?

Hah! my first thought too!


You do point out a lack of clarity in the initial problem: how many bottles of wine are you trying to save? Your solution works, with two or three bottles tossed. Unofun's works with only one bottle tossed. Mine may or may not work depending upon how immediately effective the poison is (I like his solution better than mine as it doesn't depend upon the time limit; his would work even if the wedding was 26 hours away).

 

[unofan]

Here is a question that I am curious about and I don't quite know how to solve, though I feel close.

You read the news stories about lottery winners, and think it might be fun to get in on the action.

There are five numbers drawn, from 1 to 59 inclusive, and then a sixth number, from 1 to 35 inclusive. You want to match as many numbers as you can, especially the sixth one.

The question: which gives you better odds, if either one does:
a) do you pick the same five numbers / one number each time?
b) do you allow the game computer to generate random numbers for you each time?


My intuition (which is always suspect in these situations, eh?) is that, for one week, it won't matter. But what about over the course of a year, if you play twice weekly? Would the first give slightly better odds since you are trying to match a single set of set of specific numbers to a random draw, while in the second option both your guess and the winners are changing each time?

To state it differently, I know how to calculate the odds for the first option, but I'm not sure if that calculation methodology would be valid for the second option.


Let's extend it a bit further, assuming you are going to buy, say, five tickets for each draw:
a) do you pick the same five numbers for the first set five times, and change only the sixth number for your five draws?
b) do you allow the game computer to generate randome numbers for you for each ticket?


If you match only the sixth number you get twice what you paid for the ticket. Match all six and you win millions or maybe tens of millions.

Presuming the randomized picks never repeat the same combination for a given drawing, and no other outside influences, it doesn't matter. Every combination is equally likely to provide a payout on any given drawing, and past performance is not a predictor of future results. In the long run, your expected return will be the same under any "system." A lottery is just roulette or craps writ large.

Now, in the real world where there is one outside influence - the possibility of a split jackpot due to others picking the same numbers - you are generally better off by a very very tiny amount by taking the randomized numbers because people who pick their own generally stick to lucky numbers, athlete numbers, and birthdays, all of which are predominantly under 30. A computer has no such bias and will pick numbers which are not as popular.

 

[LynahFan]

Re: Monty Hall, we have a PROBLEM

Presuming the randomized picks never repeat the same combination for a given drawing, and no other outside influences, it doesn't matter. Every combination is equally likely to provide a payout on any given drawing, and past performance is not a predictor of future results. In the long run, your expected return will be the same under any "system." A lottery is just roulette or craps writ large.

Now, in the real world where there is one outside influence - the possibility of a split jackpot due to others picking the same numbers - you are generally better off by a very very tiny amount by taking the randomized numbers because people who pick their own generally stick to lucky numbers, athlete numbers, and birthdays, all of which are predominantly under 30. A computer has no such bias and will pick numbers which are not as popular.

Concur with all of this. Even if you assumed the lottery folks rig it so that the same combination won't win twice (I doubt they bother, but I could see that picking past winning numbers is probably a strategy favored by some), then any combination that didn't win last time around (your lucky sequence OR any other random sequence) would still be equally likely in the next draw. The only way to "stack the odds" in your favor is to try to pick numbers that others don't so you don't have to split the pot.

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

Every combination is equally likely to provide a payout on any given drawing.

Right, I mentioned that for any one single given drawing, it doesn't matter, the question was: given a series of 104 drawings over the course of a year, might it matter? I also said my intuition was that it would not matter, but I wasn't sure how to prove it.

You and Lynah Fan have both reinforced my intuition with your comparable intuition, which is nice and friendly and all. I'm merely not sure whether it rises to the level of "proof" or not however.

For one drawing, completely random or selected number, the odds don't vary: 21,026,821,200 to 1 in either case to match all 6. However, over 104 drawings, might the odds be different?

I don't think so, but I'm not sure how to demonstrate that they don't.

 

[unofan]

Right, I mentioned that for any one single given drawing, it doesn't matter, the question was: given a series of 104 drawings over the course of a year, might it matter? I also said my intuition was that it would not matter, but I wasn't sure how to prove it.

You and Lynah Fan have both reinforced my intuition with your comparable intuition, which is nice and friendly and all. I'm merely not sure whether it rises to the level of "proof" or not however.

For one drawing, completely random or selected number, the odds don't vary: 21,026,821,200 to 1 in either case to match all 6. However, over 104 drawings, might the odds be different?

I don't think so, but I'm not sure how to demonstrate that they don't.

You have a roulette wheel with 38 numbers: 1-36 plus 0 & 00.
You play for 5 spins. The odds of any given number hitting on any given spin is 1/38. Whether you play the same number five times in a row or play a different number each spin, your odds of winning do not change from 1/38 on a given roll. In the long run, you will win roughly one out of every 38 spins of the wheel regardless of your betting "systems." Of course the house will only pay you at 35:1 rather than 37:1, thus ensuring you lose in the long run and it stays in business

It's not intuition, it's understanding principles of logic as applied to gambling.

 

[LynahFan]

Re: Monty Hall, we have a PROBLEM

Right, I mentioned that for any one single given drawing, it doesn't matter, the question was: given a series of 104 drawings over the course of a year, might it matter? I also said my intuition was that it would not matter, but I wasn't sure how to prove it.

You and Lynah Fan have both reinforced my intuition with your comparable intuition, which is nice and friendly and all. I'm merely not sure whether it rises to the level of "proof" or not however.

For one drawing, completely random or selected number, the odds don't vary: 21,026,821,200 to 1 in either case to match all 6. However, over 104 drawings, might the odds be different?

I don't think so, but I'm not sure how to demonstrate that they don't.

If it doesn't matter on each individual try, then it doesn't matter on 104 collective tries. 104 x 0 = 0.

Even if you know that someone has flipped 10 heads in a row with coin that you know to be "fair," the probability of a head on the next flip is still 50-50. The trials are 100% independent, so there's no way for what's gone on during past trials to affect the current one.

In the case of a lottery, you'll never be able to "prove" that the state isn't somehow manipulating the results of the current draw based on the results of past draws (or other information). Sure, the software is audited, but who audits the auditors? You can't prove a negative (i.e. that there's no "funny code" in there). However, if you take it as a given that the lottery software is fair and truly picks new, independent, random numbers each time, then your odds of winning at least once in 104 draws will be exactly the same whether you pick the same numbers each time or switch them up.

 

[unofan]

If it doesn't matter on each individual try, then it doesn't matter on 104 collective tries. 104 x 0 = 0.

Even if you know that someone has flipped 10 heads in a row with coin that you know to be "fair," the probability of a head on the next flip is still 50-50. The trials are 100% independent, so there's no way for what's gone on during past trials to affect the current one.

In the case of a lottery, you'll never be able to "prove" that the state isn't somehow manipulating the results of the current draw based on the results of past draws (or other information). Sure, the software is audited, but who audits the auditors? You can't prove a negative (i.e. that there's no "funny code" in there). However, if you take it as a given that the lottery software is fair and truly picks new, independent, random numbers each time, then your odds of winning at least once in 104 draws will be exactly the same whether you pick the same numbers each time or switch them up.

The big reason to believe that lotteries are not rigged is because, like casino games, there's no need to. The game itself is already in favor of the house based on the difference between the actual odds and the payouts. So long as it has a sufficiently large bankroll to cover short term fluctuations, the house always wins even in a straight up game.

 

[owslachief]

Re: Monty Hall, we have a PROBLEM

Hah! my first thought too!


You do point out a lack of clarity in the initial problem: how many bottles of wine are you trying to save? Your solution works, with two or three bottles tossed. Unofun's works with only one bottle tossed. Mine may or may not work depending upon how immediately effective the poison is (I like his solution better than mine as it doesn't depend upon the time limit; his would work even if the wedding was 26 hours away).

Excellent point - (1) The bride's father would like to save 12 bottles if at all possible. (2) I can narrow this to "the reception is in 25 hours" and the solution is still the same .

 

[owslachief]

Re: Monty Hall, we have a PROBLEM

Assuming that there is no problem associated with opening a bottle and then re-corking it right away, then

Answer in white:

Open four bottles and serve each of the rats enough wine from one bottle to be lethal in 24 hours. 5-1/2 hours later or so, open four more bottles and serve each of the rats wine from one of those bottles. another 5-1/2 hours later or so, open four more bottles and serve each of the rats wine from one of those bottles.

If one of the rats is dead 24 hours later, the poison was in the first batch of four. If one of the rats is dead 29-1/2 hours later, the poison was in the 2nd batch of four. If one of rats is dead 35 hours later, the poison was in the 3rd batch. If they are all alive at the start of the reception, the poison is in the unopened bottle.

I apologize because it wasn't stated clearly that the guy wants to keep the 12 good bottles'o'wine. He doesn't want to toss 4 of them - just one.

 

[owslachief]

Re: Monty Hall, we have a PROBLEM

Is the answer Jesus?

 

[joecct]

Re: Monty Hall, we have a PROBLEM

OK, you stumped me. What is that?

 

[LynahFan]

Re: Monty Hall, we have a PROBLEM

AT&T Commercial.