Making multiple bets in routlette.

Several years ago a friend, Tom, and I had a big discussion about roulette; the discussion was on a message board and the thread went for an entire page with our back-and-forth arguing. I recall I even wrote to the Wizard for his opinion and he sided with Tom but I don't remember why he said I was wrong, so I'd like to go over it again.

For the purpose of this discussion, I'm going to use a roulette wheel without any zeroes, so all you have is numbers 1 thru 36. If you put a chip on a number your odds of hitting it are 1 in 36, right?

Tom liked to bet on different individual numbers because he felt it improved his chances of hitting any one of them. I said he was betting against himself because only one number could win; to bet on two numbers (that's two individual numbers, not placing a chip on the line between two adjacent numbers) made no sense to me because you could never win on both.

His contention was that if you bet on two separate numbers your chances of winning were now 2 in 36 or 1 in 18; that betting on six numbers brought your odds down to 1 in 6. My contention was that your chances of any one number being a winner was always 1 in 36 and increasing the number of bets you made would never change that, that betting two numbers only decreased your payoff to 18-to-1 when you did win because now it cost you two chips to collect 36.

I asked him, "If you say that your chances are 1 in 18 if you bet two numbers and you pick numbers 4 and 25, which are the eighteen numbers that you have a 1 in 18 chance of hitting?" As I see it, it's impossible to answer that question.

As far as I'm concerned, no matter how many individual numbers you pick to bet on, your chances of winning are always going to be 1 in 36; even if you covered all the numbers you still have a 1 in 36 chance of winning on any one number.

I can see the strategy of betting more numbers in a game like Keno, where your odds will change according to how many numbers you match out of the total numbers you picked, but in a game like roulette, where only one number is a winner, I don't see how your odds are improved by making multiple individual bets.

Now if you were to place a bet on the line between two adjacent numbers, that's different, because now those two numbers become one entity, and your chances of winning would then indeed become 1 in 18 because you would have to assume that there are now eighteen such entities on the board: eighteen pairs of adjacent numbers.

I admit I'm not a math whiz, but for the life of me I can't see where my logic is faulty. Can someone explain in a simple way where I'm going wrong?

Yeah, you have 1-36 chance on each individual number, but your cumulative odds of hitting INCREASE with the increased coverage of numbers....and hence your expected payoff decreases by the exact amount.


Betting 1 number or betting 36 numbers doesn't change the expected value of your winning. All it does is decrease variance. So Tom, by increasing his numbers, would be decreasing his variance, but not impacting his expected outcome (in your scenario with no zeros). In fact, betting all 36 numbers would eliminate all variance and would be guaranteed to never win or lose.

I'm no math wiz either, but I think your problem (or Tom's problem) was the omission of the word 'overall'.

I'll use your theoretical 'no zero' wheel.

No matter what, the chance of winning any one number is, as you suggest, 1/36.

However, if you bet on two numbers, your overall chance of winning is 2/36, or 1/18.

If you bet all 36 numbers, you're right that the chance that any one of them hits remains 1/36, but the overall chance that any of them will hit is 36/36.

If you win, the payout is $35 regardless of how many numbers were bet.


Does that help?

Quote: DJTeddyBear

I'm no math wiz either, but I think your problem (or Tom's problem) was the omission of the word 'overall'.

I'll use your theoretical 'no zero' wheel.

No matter what, the chance of winning any one number is, as you suggest, 1/36.

However, if you bet on two numbers, your overall chance of winning is 2/36, or 1/18.

If you bet all 36 numbers, you're right that the chance that any one of them hits remains 1/36, but the overall chance that any of them will hit is 36/36.

If you win, the payout is $35 regardless of how many numbers were bet.


Does that help?

Yes, DJ, you said it better than me. Your chance of winning is 2/36....and you can easily see the 2 numbers that would win, and the 34 numbers that would not win. And I think even us non-math wizs can see that 2/36 = 1/18.

FYI: Don't think that I thought you did an inadequate job of answering.

Truth is, we were responding at the same time. I just tend to be wordy.

I think I can understand it if you stick the word "overall" in there, but it still doesn't make complete sense somehow.

I don't see how you can figure cumulative odds against a static outcome. That is to say, I don't see how you can add up the number of bets you make and use that total to figure your odds of winning if there is never more than one winning number drawn per game.

Quote: tsmith

I don't see how you can figure cumulative odds against a static outcome. That is to say, I don't see how you can add up the number of bets you make and use that total to figure your odds of winning if there is never more than one winning number drawn per game.

Yes. Individually, they each have a 1/36 chance (still thinking about this hypothetical 'no zero' wheel.)

But overall, don't you think you have double the odds if you make two different bets?

The math is simple:

'odds of winning' equals 'number of bets' divided by 'number of possible outcomes'

So with two bets, the odds are 2/36.

Oh, sure, you will ALWAYS lose one of those bets. But the odds of winning one, is the combined odds of hitting either of them.

The guaranteed loss does not affect the odds. The extra chance to win does.

Quote: DJTeddyBear

But overall, don't you think you have double the odds if you make two different bets?

See, that's where I get confused. I look at that as betting against yourself. In effect you're saying "I think number 12 is going to win AND I think number 43 is going to win," and that can never happen.

I suppose I'm stuck on thinking in terms of "and" instead of "or".

To clarify more the case I can add a simple mat explain.
Consider each number bet as individual on the table, but as the number rolled is the same
you have 1/36 of wining for each number you place a bet on.
As this bets are not dependent each one, the global probability of an event (or a combination of not depenedent events),
its equal to the sum of the probabilities of each event happening. Having this, you now have
that for 2 numbers the probability of winning is P(win) = 1/36 + 1/36 which is what DJTeddy and the others are trying to explain.

I might have confuse the whole idea, because my english isn't to good, but I expect you understand this.

Quote: tsmith

See, that's where I get confused. I look at that as betting against yourself. In effect you're saying "I think number 12 is going to win AND I think number 43 is going to win," and that can never happen.

I suppose I'm stuck on thinking in terms of "and" instead of "or".

Yep, that's exactly where you are getting confused. Would you not agree that if you bet every number you would be guaranteed to win on one of the bets? If so, it should make sense to you that the more numbers you bet the greater your odds of hitting ONE of your numbers.

Again, betting 1, or 2, or 12 or 36 numbers doesn't change your "odds"...you can't increase or decrease the odds. But you can reduce variance, which is what Tom is attempting to do - even if he doesn't realize it.

The way I look at it, if you want to bet on more than one number to increase your chances of winning, it makes more sense to put the chip on a line instead of an individual number. That way, as I said, more than one number is included as a single entity and then I can understand it as an "or" situation.

Sure, if either of the numbers wins you might think you're ending up with a smaller payoff, but you're really getting exactly the same thing, aren't you? 36/1 for one number or 18/1 for two numbers.

If there are 36 beautiful women in a bar, your approach to one of them will affect your chances with the others.
But the numbers on a roulette wheel do not react emotionally to your betting on other numbers. If there are 36 slots on the wheel that little ball will definitely land in one of them and each slot has a chance of 1 in 36 irrespective of whether you make zero bets, one bet, two bets or whatever.

The only thing that changes is that since you have made two straight-up bets, you MUST lose one of them and MIGHT lose the other also.

You are not betting against yourself with that second bet. You are merely acknowledging that while emotionally you want that first bet to win, you are no more certain that it will win than any other number. Your first bet has a 1 in 36 chance in winning. Your second bet has a 1 in 36 chance of winning. Its no more "betting against yourself" than if some other player comes up and places a bet on a different number than you chose. That other player is not betting against you. He is betting against the casino. And so are you.

Betting against yourself? Not really.

Let's say you and your spouse each make one bet on that roulette table, and one of you wins. Sure, whoever won will have bragging rights, but, as a couple, didn't you both win?

Quote: tsmith

The way I look at it, if you want to bet on more than one number to increase your chances of winning, it makes more sense to put the chip on a line instead of an individual number. That way, as I said, more than one number is included as a single entity and then I can understand it as an "or" situation.

Sure, if either of the numbers wins you might think you're ending up with a smaller payoff, but you're really getting exactly the same thing, aren't you? 36/1 for one number or 18/1 for two numbers.

I take it this means you are OK with placing a chip on the line between two numbers. From there, consider doubling your bet and placing two chips on the line between the two numbers. From there, recognize that this is exactly the same thing as placing one chip on each of those two numbers. Finally, consider the possibility that the two numbers that interest you are not adjacent to each other and the only way to cover them is to put a single chip on each, since you can't put the two chips on the line between. Does that help?

Do you feel that betting "Red" is betting against yourself? How about if you increased your "red" bet to 18 chips? How about placing a chip on each of the 18 red numbers? At what point does it seem to you that you have started betting against yourself?

Actually, you are betting against yourself, but not in a way that reduces your expected payout. In each case in the previous paragraph (for no zeros) your expected win is half the amount you bet and your expected (average) outcome is the same as your starting point. Half the time you will lose your bet; half the time you will double your money.

Quote: tsmith

"I think number 12 is going to win AND I think number 43 is going to win," and that can never happen.

Well, of course it can't happen. There are only 36 numbers, and thus 43 could never be a winning number. :)

I think a lot of confusion comes into play with many non-gamblers / novice gamblers in the same thing that you mentioned of the "overall".

In a perfect non-biased 36-slot wheel, if the wheel is spun 36 times, then each number would come up once in those 36 times. Thus, if you spin the wheel 36 times, and you've bet on 12 and 34, then the 12 would result once, and the 34 would result once. Thus, you have won 2 out of the 36 times, or 2/36, which simplified in mathematical terms, comes to 1/18, which is the 1 of 18 that people will quote to you as the odds of winning on this system.

But, if you aren't looking at the overall, then you are right in that the concept of betting on two separate numbers doesn't make much sense. I, for one, don't enjoy roulette all that much. That being said, I've been known to take the extra money left over after a bad run of losses at the craps table and/or PaiGow Poker table, and place it on one number, one spin, at the roulette wheel. In this example, I don't particularly see any reason to divide up my $5 between two separate numbers. I only care about one spin, and in that case, I only care about one number.

So, when your friend is talking about betting two numbers and having a 1 in 18 chance of winning, what he's really saying is that by betting two numbers, over the long run (overall), over multiple spins, whether they be at one sitting or multiple sittings, that 1 out of every 18 spins will result in a win for him. But, if you're only going to be concerned with one spin, then yes, you have a 1 out of 36 chance of hitting your number.

Quote: FleaStiff

If there are 36 beautiful women in a bar,

Please tell me where this bar is. :)

Quote: tsmith

The way I look at it, if you want to bet on more than one number to increase your chances of winning, it makes more sense to put the chip on a line instead of an individual number. That way, as I said, more than one number is included as a single entity and then I can understand it as an "or" situation.

Sure, if either of the numbers wins you might think you're ending up with a smaller payoff, but you're really getting exactly the same thing, aren't you? 36/1 for one number or 18/1 for two numbers.

As someone else said, betting $10 on the line is the SAME EXACT THING as betting $5 on each of the two numbers. And the payout is also the exact same. In fact, on your fictional 36 slot wheel, EVERY bet has the same exact expected value. Bet the corner, bet Red, bet Even, it doesn't matter.

Quote: tsmith

The way I look at it, if you want to bet on more than one number to increase your chances of winning, it makes more sense to put the chip on a line instead of an individual number. That way, as I said, more than one number is included as a single entity and then I can understand it as an "or" situation.

Sure, if either of the numbers wins you might think you're ending up with a smaller payoff, but you're really getting exactly the same thing, aren't you? 36/1 for one number or 18/1 for two numbers.

As someone else said, betting $10 on the line is the SAME EXACT THING as betting $5 on each of the two numbers. And the payout is also the exact same. In fact, EVERY bet has the same exact expected value. Bet the corner, bet Red, bet Even, it doesn't matter.

Quote: konceptum

Well, of course it can't happen. There are only 36 numbers, and thus 43 could never be a winning number. :)

Oops, I meant to say 34, not 43.

Quote:

I only care about one spin, and in that case, I only care about one number.

So, when your friend is talking about betting two numbers and having a 1 in 18 chance of winning, what he's really saying is that by betting two numbers, over the long run (overall), over multiple spins, whether they be at one sitting or multiple sittings, that 1 out of every 18 spins will result in a win for him. But, if you're only going to be concerned with one spin, then yes, you have a 1 out of 36 chance of hitting your number.

Okay, I think I'm catching on now.

When I think about Roulette I think in terms of one game (spin) at a time. I think I can understand now what Tom would be doing by making multiple bets per game, because he would be concerned with the total number of games (spins) he'd be playing.

Thanks for cutting thru the fog.