Word Problem: Three Roulette tables. Any and all help appreciated.

I’d appreciate if someone could give solving this problem a go. The best analogy I could come up with was using Roulette tables. I’m not sure if the problem even has a solution.

Imagine that there are 3 roulette tables side by side. All three are spun simultaneously. You can bet on all three. You can bet as many times as you like.

- If the balls of table 1 and table 2 land on Red, and the ball of table 3 lands on Black, you profit.
- If the balls of table 1 and table 2 land on Black, and the ball of table 3 lands on Red, you profit.
- If the balls of table 1 and 2 land on Red, and the ball of table 3 also lands on Red, you lose.
- If the balls of table 1 and 2 land on Black, and the ball of table 3 also lands on Black, you lose.
- If the balls of table 1 and 2 land on different colours, you beakeven (irrespective of table 3’s colour).

Is there any way of betting which would make this scenario possible? Any progression that can be used? You can continue betting for as long as you like, provided that by the time you leave the casino, your bank balance reflects the profit or loss which would be produced by the above rules.

Thanks.

Re: Word Problem: Three Roulette tables. Any and all help appreciated.

From what I recall, a roulette table has 60 spaces and I think two of them are green (neither red nor black). Perhaps fair coins would work better. You flip three fair coins, and if the first two are the same, but the third is different, you profit. If all three are the same, you lose, and if the first two are different, you break even. I am not sure what you mean by "You can bet on all three". The rules you lay out only describe what happens from the "joint" outcome, not what happens at each roulette table individually.

Since all three coins are fair, outcomes will be one of the following eight:

$\displaystyle \begin{array}{ccc}\text{Outcome} & W/L/D & \text{Probability} \\ HHH & L & 0.125 \\ HHT & W & 0.125 \\ HTH & D & 0.125 \\ HTT & D & 0.125 \\ THH & D & 0.125 \\ THT & D & 0.125 \\ TTH & W & 0.125 \\ TTT & L & 0.125\end{array}$

So, the probability of winning is $\displaystyle 0.125 + 0.125 = 0.25$, the probability of losing is $\displaystyle 0.125 + 0.125 = 0.25$ and the probability of breaking even is $\displaystyle 4\cdot 0.125 = 0.5$. Hence, the game is a fair game. By the law of large numbers, there is 100% chance that if you have infinite funds (or the casino is willing to float you infinite credit), you can achieve any amount of winnings that you want. But, in general, you have even chances of winning or losing in this scenario.

Now, if you want to include the rules for Roulette, as well, and these additional methods of winning are "side bets", that changes the problem drastically.

Re: Word Problem: Three Roulette tables. Any and all help appreciated.

Thanks for your reply.

It’s quite a strange question I know. I probably should have used coins like you did but it didn’t occur to me. I understand that it’s a fair game, that’s actually beneficial to me (bear in mind that this problem isn’t actually about roulette).

If you were to flip three coins one after the other then you could bet according to the previous one. For example: you see the first coin being flipped and its heads; you then bet that the second coin will be heads and that the third will be tails.

But the challenge of my question and why I’m struggling with it so much is because the coins aren’t flipped in succession. They’re flipped simultaneously. You have to bet before the three coins are flipped.

Re: Word Problem: Three Roulette tables. Any and all help appreciated.

Right, I made the assumption that the three coins were flipped simultaneously. The probabilities I got were that the game is fair. So, the question becomes, how much "profit" do you receive when you profit? If you bet $10, do you win $10? How about if you lose? If you bet $10, and lose, do you lose $10? If that is the case, then as I stated, given infinite credit and an infinite amount of time, if you set a goal for your winnings (like you will stop when you have $n), the law of large numbers guarantees that you will achieve your goal. If you have a finite amount of money to start, then you can probably figure out the optimal bet to maximize your chances of eventually getting $n.

Re: Word Problem: Three Roulette tables. Any and all help appreciated.

Yes, if you bet $10 and win, you win $10. If you lose, you lose $10.

The coins aren’t fair. Coins 1 & 2 have a strong tendency to land the same way up, and a strong tendency to land the opposite way up to coin 3.

Betting in a single direction won’t get you anywhere because HHT is as likely as TTH.

However, 80% of the time coin 1 & 2 are the same, and different to coin 3. My question is how you take advantage of this fact.

Re: Word Problem: Three Roulette tables. Any and all help appreciated.

Ok, you need to give probabilities for all outcomes:

P(HHH) = ?

P(HHT) = 40%

P(HTH) = ?

P(HTT) = ?

P(THH) = ?

P(THT) = ?

P(TTH) = 40%

P(TTT) = ?

Note, the probabilities must add up to 100%. From your rules, you win if EITHER outcome HHT or TTH comes up, but lose if EITHER HHH or TTT comes up. Now, are you saying that you can only bet on HHT or TTH? So, if you bet HHT, you have a 40% chance of winning, but if HHH, TTT, or TTH comes up, you lose? If that is the case, the game is no longer fair, and in the long run, you are guaranteed to lose money.

Re: Word Problem: Three Roulette tables. Any and all help appreciated.

The exact probabilities don’t really matter. The other probabilities could all be 5% (to add up to the 100%). You can bet any amount. Obviously as you say, if you bet only HHT or TTH, you would only win 40% and in the long run lose. You have freedom to bet how you like.

It is not a fair game in the sense that the game has bias: HHT and TTH happen the majority of the time. The question is whether there is a way of creatively gambling to take advantage of this bias.

Re: Word Problem: Three Roulette tables. Any and all help appreciated.

The same question can be phrased in terms of the stock market. You have three stocks.

Each stock can move up or down.

80% of the time when stock 3 goes down, stock 1 and 2 rise.

80% of the time when stock 3 goes up, stock 1 and 2 go down.

You can buy or sell any of the three stocks, with any amount you like. How do you take advantage of this information?

Re: Word Problem: Three Roulette tables. Any and all help appreciated.

Now that changes things significantly! Suppose if stock 1 goes up, you will get $a, and if it goes down, you will lose $a. If stock 2 goes up, you get $b, down means you lose $b, and if 3 goes up, you get $c, but if it goes down, you lose $c. Assuming stock 3 has 50% chance of going up and 50% chance of going down, let's figure out the probable outcomes:

$\displaystyle \begin{array}{cc}\text{Outcome} & \text{Probability} \\ UUU & 0.5p_1 \\ UUD & 0.5\cdot 0.8 \\ UDU & 0.5p_2 \\ UDD & 0.5q_1 \\ DUU & 0.5 (0.2 - p_1 - p_2) \\ DUD & 0.5q_2 \\ DDU & 0.5\cdot 0.8 \\ DDD & 0.5(0.2 - q_1 - q_2)\end{array}$

So, your money after the "outcomes" will be

$\displaystyle 0.5p_1(a+b+c) + 0.4(a+b-c) + 0.5p_2(a+c-b)+0.5q_1(a-b-c)+0.5(0.2-p_1-p_2)(b+c-a)+0.5q_2(b-a-c) + 0.4(c-a-b) + 0.5(0.2-q_1-q_2)(-a-b-c)$

Collecting all terms, we get:

$\displaystyle 0.5(p_1+0.8 + p_2 + q_1 -0.2+p_1+p_2-q_2-0.8-0.2+q_1+q_2)a = (p_1+p_2+q_1-0.2)a$

$\displaystyle 0.5(p_1+0.8-p_2-q_1+0.2-p_1-p_2+q_2-0.8-0.2+q_1+q_2)b = (q_2-p_2)b$

$\displaystyle 0.5(p_1-0.8+p_2-q_1+0.2-p_1-p_2-q_2+0.8-0.2+q_1+q_2)c = 0$

So, buying stock 3 will allow you to break even. If $\displaystyle p_1+p_2+q_1-0.2>0$, buying stock 1 is beneficial. If $\displaystyle q_2-p_2>0$, buying stock 2 is beneficial. Unless I have all probabilities, I cannot solve this problem further.

Re: Word Problem: Three Roulette tables. Any and all help appreciated.

I hope I followed your work correctly. Is there not a problem that you’re relying on the assumption that stock 3 has a 50% chance of going up? What if it isn’t 50:50? Even if it is, I'm not sure that helps us make money.

Here is a simple example:

S1 S2 S3

U U D

D D U

The market did what we expected:the majority (in this case 100%) of moves by stock3 saw stocks 1&2 moving together in the opposite direction. However, if we were to buy stocks 1&2, we would have only broken even.

It may be that, as you say, without more information it is not possible to create a strategy to make money. I will spend some more time trying to look for a progression, or similar to try to take advantage of the bias. Thank you for your time, and if you have any ideas please do share.

Re: Word Problem: Three Roulette tables. Any and all help appreciated.

There are wayyyy too many variables in this system. For example, typically, if you buy a stock, you don't earn your entire investment if it goes up, and similarly, you don't lose your entire investment if it goes down. What happens if the stock stays pretty much the same price? The "what if"s are pretty much endless. That is why analysts spend so much time studying the stock market. It is an extremely complex question. We try to make the system as simple as possible, but some guessing is always involved.