Reducing the Numbers, Increasing the Odds?
July 9, 2017 9:27 AM Subscribe
Do my odds at winning any game of chance increase if numbers that have already won are removed from the list of possibilities? Its always said that every number always has an equal chance of randomly coming up.
But it just SEEMS like a number that has already come up is less likely to come up soon as a number that has never. In lotteries with hundreds of millions of winning combinations, this may be insignificant wishful thinking. But since casinos also win by tenths or hundredths of percent over random chance, I was just wondering. States follow this closely and I rarely hear of the same set of numbers winning twice in the history of a lottery.
But it just SEEMS like a number that has already come up is less likely to come up soon as a number that has never. In lotteries with hundreds of millions of winning combinations, this may be insignificant wishful thinking. But since casinos also win by tenths or hundredths of percent over random chance, I was just wondering. States follow this closely and I rarely hear of the same set of numbers winning twice in the history of a lottery.
In Blackjack, it's the reverse. The fewer Aces and honors that have been played, the better your odds of being dealt 20 or 21. Counting cards is big in Blackjack.
posted by SemiSalt at 9:41 AM on July 9, 2017
posted by SemiSalt at 9:41 AM on July 9, 2017
Best answer: Do my odds at winning any game of chance increase if numbers that have already won are removed from the list of possibilities? Its always said that every number always has an equal chance of randomly coming up.
There is a difference between 'removed from possibility' and 'has happened before'.
REMOVE FROM POSSIBILITY:
So, this is the basis of card counting. Let's say you have a basic 52 card deck, and, most importantly, you don't shuffle every hand but let played cards rest by the side of the table until you run out of cards. In this situation, if the 2 of clubs has been played, there's no other 2 of clubs there.
If you know that lots of 'bad cards' have been played, and very few good cards have been played, you might start betting heavily on the hopes you'll get a good hand. (This is card counting and will get you blacklisted from any casino and may be illegal in your jurisdiction, so, y'know: don't.)
This is why casinos tend to play with 6- to 8-deck 'shoes' (a container which holds many decks, all shuffled together): it makes card counting less reliable and/or less worth your time, and reduces the time spent shuffling (because casinos loathe moments where people are sitting doing nothing). This results in a better 'house edge' (chance that the casino will win). If they wanted to basically kill card counting at that table, they will switch to shuffling after every hand.
HAS HAPPENED BEFORE:
Let's say you're doing one of those lottos where you draw balls from those big transparent spinning machines. They don't permanently remove balls; they remove balls for as long as that draw takes, and then put them back in the spinning machine. The odds of the same number coming up twice are the same because the set of possible draws remains the same.
I mean, someone drew lucky numbers for a Chinese fortune cookie, and then later the lotto drew the same numbers...
Why hasn't a lotto had identical winning numbers before? Because very large numbers/many combinations of numbers means it will take a very long time to go through every possibility. That is why it is stunningly unlikely that your lotto numbers will match the draw; it's the exact same math* to determine that the draw's chance of matching a previous draw is unlikely. And because there are fewer draws (1 to 7 a week per lotto, usually) than people playing the lotto (however many people buy a ticket), a lotto draw-lotto draw match will theoretically take much longer to happen.
*Minus the fact that people, unlike the draws, are not good at being random.
posted by flibbertigibbet at 9:53 AM on July 9, 2017 [9 favorites]
There is a difference between 'removed from possibility' and 'has happened before'.
REMOVE FROM POSSIBILITY:
So, this is the basis of card counting. Let's say you have a basic 52 card deck, and, most importantly, you don't shuffle every hand but let played cards rest by the side of the table until you run out of cards. In this situation, if the 2 of clubs has been played, there's no other 2 of clubs there.
If you know that lots of 'bad cards' have been played, and very few good cards have been played, you might start betting heavily on the hopes you'll get a good hand. (This is card counting and will get you blacklisted from any casino and may be illegal in your jurisdiction, so, y'know: don't.)
This is why casinos tend to play with 6- to 8-deck 'shoes' (a container which holds many decks, all shuffled together): it makes card counting less reliable and/or less worth your time, and reduces the time spent shuffling (because casinos loathe moments where people are sitting doing nothing). This results in a better 'house edge' (chance that the casino will win). If they wanted to basically kill card counting at that table, they will switch to shuffling after every hand.
HAS HAPPENED BEFORE:
Let's say you're doing one of those lottos where you draw balls from those big transparent spinning machines. They don't permanently remove balls; they remove balls for as long as that draw takes, and then put them back in the spinning machine. The odds of the same number coming up twice are the same because the set of possible draws remains the same.
I mean, someone drew lucky numbers for a Chinese fortune cookie, and then later the lotto drew the same numbers...
Why hasn't a lotto had identical winning numbers before? Because very large numbers/many combinations of numbers means it will take a very long time to go through every possibility. That is why it is stunningly unlikely that your lotto numbers will match the draw; it's the exact same math* to determine that the draw's chance of matching a previous draw is unlikely. And because there are fewer draws (1 to 7 a week per lotto, usually) than people playing the lotto (however many people buy a ticket), a lotto draw-lotto draw match will theoretically take much longer to happen.
*Minus the fact that people, unlike the draws, are not good at being random.
posted by flibbertigibbet at 9:53 AM on July 9, 2017 [9 favorites]
flibbertigibbet has a great point about the difference between removal and duplication, which it's not clear from your question which one you are truly interested in.
If you guarantee that past willing numbers never appear, then the benefit depends on the original odds. Consider a dice roll; if you roll a die, then somehow guarantee that the number rolled will not appear again, your odds of picking the rolled number go from 1 in 6 to 1 in 5. Not a guarantee, but noticeably better. But consider the lottery, which is about 30 years old. (Powerball started as Lotto*America in 1988.) For simplicity, let's say the odds have always been 1 in 100 million. If you take 30 years x 52 weeks x 2 drawings per week, you have 3120 drawings over its history. If you eliminate those numbers from the next draw, you still have odds of 1 in 99,996,880 - technically better odds, but realistically it's an unnoticeable difference.
Repeat numbers do seem to be unlikely, but the classic example is that if you flip a coin 7 times and it comes up heads every time, the odds of heads on the 8th flip are still 50%. Another way to look at that which helped me: The odds of flipping heads 8 times in a row are 1 in 256. So it seems really unlikely that you'd get 8 in a row. But the odds of 7 heads in a row are 1 in 128, and in our example, that has already happened so you can eliminate those long odds because we've already overcome them.
I dealt blackjack many years ago and saw people bet 'negative progression' based on this fallacy. You start by betting one unit (say, $5). If you win any hand, you reset your bet to $5, but you double your bet each time you lose. The idea is that eventually you will win a hand because, well, you HAVE to ("I'm due!"), and every time you do you'll be one unit farther ahead. People thought this was a sure bet. Until they lost their 6th hand in a row having bet $160, and realized they couldn't double their bet again because the table limit was only $200. Now in the span of 90 seconds they're $315 in the hole and their strategy is destroyed.
posted by SquidLips at 10:31 AM on July 9, 2017 [2 favorites]
If you guarantee that past willing numbers never appear, then the benefit depends on the original odds. Consider a dice roll; if you roll a die, then somehow guarantee that the number rolled will not appear again, your odds of picking the rolled number go from 1 in 6 to 1 in 5. Not a guarantee, but noticeably better. But consider the lottery, which is about 30 years old. (Powerball started as Lotto*America in 1988.) For simplicity, let's say the odds have always been 1 in 100 million. If you take 30 years x 52 weeks x 2 drawings per week, you have 3120 drawings over its history. If you eliminate those numbers from the next draw, you still have odds of 1 in 99,996,880 - technically better odds, but realistically it's an unnoticeable difference.
Repeat numbers do seem to be unlikely, but the classic example is that if you flip a coin 7 times and it comes up heads every time, the odds of heads on the 8th flip are still 50%. Another way to look at that which helped me: The odds of flipping heads 8 times in a row are 1 in 256. So it seems really unlikely that you'd get 8 in a row. But the odds of 7 heads in a row are 1 in 128, and in our example, that has already happened so you can eliminate those long odds because we've already overcome them.
I dealt blackjack many years ago and saw people bet 'negative progression' based on this fallacy. You start by betting one unit (say, $5). If you win any hand, you reset your bet to $5, but you double your bet each time you lose. The idea is that eventually you will win a hand because, well, you HAVE to ("I'm due!"), and every time you do you'll be one unit farther ahead. People thought this was a sure bet. Until they lost their 6th hand in a row having bet $160, and realized they couldn't double their bet again because the table limit was only $200. Now in the span of 90 seconds they're $315 in the hole and their strategy is destroyed.
posted by SquidLips at 10:31 AM on July 9, 2017 [2 favorites]
Squid--that's the entire point of table limits in blackjack, right?
posted by radicalawyer at 11:12 AM on July 9, 2017
posted by radicalawyer at 11:12 AM on July 9, 2017
Squid--that's the entire point of table limits in blackjack, right?
I wouldn't think the only point. Martingale betting "strategies" don't actually work even on no limit tables. Why should casinos care? Even if you have infinite money, the strategy doesn't harm the casino, because every single hand or spin played has the same vigorish as every other one. From the perspective of the casino, you're just someone placing increasingly large bets and, on average, losing money. The fact that from your perspective you would probably manage to end up without losing your money during any particular session doesn't change the fact that, in both the short and long run, you will, on average, lose money.
There are some wholly theoretical situations where Martingale systems could be described as winning strategies, but these have nothing to do with any gambling that takes place in the real world.
posted by howfar at 11:28 AM on July 9, 2017 [1 favorite]
I wouldn't think the only point. Martingale betting "strategies" don't actually work even on no limit tables. Why should casinos care? Even if you have infinite money, the strategy doesn't harm the casino, because every single hand or spin played has the same vigorish as every other one. From the perspective of the casino, you're just someone placing increasingly large bets and, on average, losing money. The fact that from your perspective you would probably manage to end up without losing your money during any particular session doesn't change the fact that, in both the short and long run, you will, on average, lose money.
There are some wholly theoretical situations where Martingale systems could be described as winning strategies, but these have nothing to do with any gambling that takes place in the real world.
posted by howfar at 11:28 AM on July 9, 2017 [1 favorite]
Best answer: Consider flipping a coin. There are only two options, heads or tails. Each has a 50% probability. If you flip heads on your first flip, by your thinking, it would be more likely to flip tails on your second. But how is that possible? Assuming the coin isn't weighted, there are still only two options, and both are still equally likely. How could those two options be anything other than 50-50?
Let's next consider flipping two coins. There are four options here: HH, HT, TH, TT. You flip heads on the first flip. That leaves two options for the total outcome, HH and HT. Again, the probability is 50% for each option, the same as it was for the first flip.
Now, if you remove the option of heads after the first flip, that changes things significantly. The only option is tails, and the outcome is 100% going to be HT.
This is why flibbertigibbet drew the distinction between the two: one does affect probability, and one does not.
posted by kevinbelt at 11:37 AM on July 9, 2017
Let's next consider flipping two coins. There are four options here: HH, HT, TH, TT. You flip heads on the first flip. That leaves two options for the total outcome, HH and HT. Again, the probability is 50% for each option, the same as it was for the first flip.
Now, if you remove the option of heads after the first flip, that changes things significantly. The only option is tails, and the outcome is 100% going to be HT.
This is why flibbertigibbet drew the distinction between the two: one does affect probability, and one does not.
posted by kevinbelt at 11:37 AM on July 9, 2017
To clarify why they're relevant to this kind of intuitive difficulty, Martingale systems don't work for exactly the same reason that picking old winning numbers is no worse than picking any other set of numbers. Each random event is wholly independent of every other. The world doesn't know, or care, when you make the individual choice to make a bet, that there were a dozen bets before it. The fact that the probability of 8 heads in a row is 1/256 doesn't mean that the bet of on the eight coin toss coming up tails has odds of 255/256 - you are still betting 128 betting units on a coin toss, and have a 50% chance* of losing that, along with everything else you've already lost. The intuition that it's a potentially winning technique comes from our resistance to viewing probability in the independent terms that this particular sort of analysis requires.
*In a casino, of course, the average return on any hand, throw etc is always a bit worse than 1.
posted by howfar at 11:40 AM on July 9, 2017
*In a casino, of course, the average return on any hand, throw etc is always a bit worse than 1.
posted by howfar at 11:40 AM on July 9, 2017
1-2-3-4-5-6 is absolutely as likely as any other 6 ball draw combination in a lotto. The fact that you intuition tells you not to bet that way is a good sign of how unlikely it actually is to win.
posted by MattD at 12:10 PM on July 9, 2017 [1 favorite]
posted by MattD at 12:10 PM on July 9, 2017 [1 favorite]
The effect of card removal in single deck blackjack is real enough that it's basically impossible to find a single deck table in Las Vegas that still pays 3:2 on blackjack. You can find single deck tables but they pay 6:5 (or worse), which gives a huge house edge. Never play blackjack if it pays less than 3:2.
In multi-deck blackjack the effect of card removal is pretty small (here it is for a six deck shoe), and even that effect is removed by the use of a continuous shuffling machine. It turns out the house edge is actually slightly smaller with a CSM than without one. On the other hand the use of a CSM increases the number of hands dealt in an hour, so that smaller house edge is literally made up for in volume (see the last paragraph of the above link). I suggest reading around the Wizard of Odds site if you really want to get into the weeds of cut card effects and shoes vs CSMs vs shuffling every hand, and so on.
Squid--that's the entire point of table limits in blackjack, right?
The table limit is a function of the minimum bet, and the house uses the minimum bet and the house edge for a particular table to figure out how much money they're going to make from that table. If you want to bet more than the maximum for a particular table, they'll encourage you to play at another table with a higher limit (which also has a higher minimum bet) so they still make their numbers on you. If you play at a low limit table you won't get as many free drinks and they might not be happy about rating you for comps. Play at a higher limit table with a higher minimum bet, and the house will treat you better. But that has nothing to do with the effect of removing cards from the deck.
posted by fedward at 12:57 PM on July 9, 2017
In multi-deck blackjack the effect of card removal is pretty small (here it is for a six deck shoe), and even that effect is removed by the use of a continuous shuffling machine. It turns out the house edge is actually slightly smaller with a CSM than without one. On the other hand the use of a CSM increases the number of hands dealt in an hour, so that smaller house edge is literally made up for in volume (see the last paragraph of the above link). I suggest reading around the Wizard of Odds site if you really want to get into the weeds of cut card effects and shoes vs CSMs vs shuffling every hand, and so on.
Squid--that's the entire point of table limits in blackjack, right?
The table limit is a function of the minimum bet, and the house uses the minimum bet and the house edge for a particular table to figure out how much money they're going to make from that table. If you want to bet more than the maximum for a particular table, they'll encourage you to play at another table with a higher limit (which also has a higher minimum bet) so they still make their numbers on you. If you play at a low limit table you won't get as many free drinks and they might not be happy about rating you for comps. Play at a higher limit table with a higher minimum bet, and the house will treat you better. But that has nothing to do with the effect of removing cards from the deck.
posted by fedward at 12:57 PM on July 9, 2017
In a truly random drawing, all numbers have the same chance of coming up (even if they've come up before).
However, achieving true randomness is harder than you'd think, especially when there are physical components at work. There was a gambling syndicate that scouted out all the roulette wheels in a casino, looking for specific wheels that hit a few numbers more often than would be expected. This was enough to overcome the house advantage, as long as they found those tables (the casino started moving them around) and wagered only on the more-likely numbers.
So under some circumstances, it lowers your odds to exclude previously-selected numbers. Now, the people who run these drawings know that everyone is looking for an edge, and work to keep things equal and as random as possible. That said, there are still exploitable quirks in the logistics of lotteries that can allow people to win more often that you'd expect.
posted by Huffy Puffy at 2:26 PM on July 9, 2017
However, achieving true randomness is harder than you'd think, especially when there are physical components at work. There was a gambling syndicate that scouted out all the roulette wheels in a casino, looking for specific wheels that hit a few numbers more often than would be expected. This was enough to overcome the house advantage, as long as they found those tables (the casino started moving them around) and wagered only on the more-likely numbers.
So under some circumstances, it lowers your odds to exclude previously-selected numbers. Now, the people who run these drawings know that everyone is looking for an edge, and work to keep things equal and as random as possible. That said, there are still exploitable quirks in the logistics of lotteries that can allow people to win more often that you'd expect.
posted by Huffy Puffy at 2:26 PM on July 9, 2017
This thread is closed to new comments.
Checking my local/provincial lotto rules (match 6 numbers from 01 to 99) .
There are more than 800billion possible combinations
If we had a drawing every second or every day it would still take more than 25 thousand years to enumerate them all.
posted by mce at 9:36 AM on July 9, 2017 [1 favorite]