Help me untangle a mental knot

bobcat

bobcat

Well-known Member
Location
Northern Calif
I was having a discussion with AI about blackjack. As we all know the odds favor the dealer. Out of a hundred hands the average would be the dealer wins about 48, and the player wins about 44, and about 8 pushes. Granted that's just an average, and streaks happen. Now here is the conundrum:

The likelihood of the dealer winning all 100 hands (Given normal betting practices) is next to zero (Not impossible, but almost). AI agrees with this. As the winning streak continues beyond the normal 48, the unlikeliness of winning the next hand in the streak increases. AI agrees with this. However, even if the dealer has won 99 out of 100 hands, the odds of winning the next hand are still a constant in the dealer's favor at 48% as opposed to 44%. AI says this is true.

According to what it tells me, each hand is totally independent of the one before it (Assuming one isn't counting cards and making strategic bets).
On the other hand, streaks are not independent. The longer they continue, the more unlikely they become.

The problem is I can't reconcile those two seeming contradictions, but I am assured they are both true. If the dealer has won 90 hands in a row, and it is extremely unlikely that streak would continue with the next hand, then how are the odds not changed for that next hand?
 
What if is like an Optical illusion. We see it as the universe can not keep this up, but the odds do not change on the next deal...it will always be 48%. Other wise you could calculate the odds as he keeps winning. But we can not find the mathematical formula to figure it. ?
 
bobcat said:
The problem is I can't reconcile those two seeming contradictions, but I am assured they are both true. If the dealer has won 90 hands in a row, and it is extremely unlikely that streak would continue with the next hand, then how are the odds not changed for that next hand?
Click to expand...

Well, you got my curiosity up. I had to ask AI why the dealer has the advantage. AI says...

The dealer has a mathematical advantage in blackjack simply because players must act first. If you bust (exceed 21), you lose your bet immediately. If the dealer busts later in the same round, the casino still keeps your money, effectively taking advantage of the "double-bust" scenario. [1, 2, 3]
Click to expand...

Paco Dennis said:
What if is like an Optical illusion. We see it as the universe can not keep this up, but the odds do not change on the next deal...it will always be 48%. Other wise you could calculate the odds as he keeps winning. But we can not find the mathematical formula to figure it. ?
Click to expand...
You are right. Every time the cards are dealt, the odds are the same, because nothing changes. You would need to be the one who plays last to change the odds of you winning.
 
I am watching a weather report and thinking playing Black Jack is not like figuring where and when a severe flash flood will hit. The odds of being hit by a flash flood change as the conditions change. The deal of Black Jack does not change, no matter how many times it is repeated.
 
The problem for me is reconciling the relationship between the two seeming contradictions.
If it's true that the dealer winning streak becomes more unlikely the longer it continues: Example - If the dealer just won their first hand, it's not that unlikely to win the next one (Two in a row). However, if the dealer has won 60 in a row, then the likelihood of winning the next one becomes highly unlikely. That makes sense to me.

However, each hand played, no matter if the dealer has won 60 in a row, the chance of them winning the next one remains as constant as if they had won just two in a row, because each hand is independent of the last one.
Those two concepts don't come together in my mind. It seems that if the streak becomes more unlikely the longer it continues, then the odds should change for the next hand. But they don't.
 
Dybbuk said:
Check out the Poisson Distributions. It's all about probabilities.
Click to expand...
I checked it out, and it echoes what I have been saying about probabilities. Here's another example:
Lets say the Atlantic coast has an average of 5 hurricanes per year. It is highly unlikely that they would only have 1 in a given year. It is also extremely unlikely that they would have 20 in a given year. However, the weather doesn't know or keep track of how many it's had so even if it's had 19 the odds of it having another one is no different than if there has been only 1.

It just seems to me that if something becomes increasingly unlikely, then by definition, the odds of it happening again should be affected, but that doesn't happen.
 
Hmmmm. Interesting. Let's use another scenario. You are betting whether a single card pulled out of the deck will be red or black (50/50 odds). All 52 cards are shuffled after each draw. The last 20 draws have been black. What would the 'vegas' odds be that the next card would be black? I think the vegas odds would be 50/50. In other words, the likelihood that the streak continues is 50/50. No ?

I agree, it's a head scratcher. After 20 black cards in a row, you think there is 'pressure' from somewhere to make a red card show up, lol.
 
Last edited:
RandomName said:
Hmmmm. Interesting. Let's use another scenario. You are betting whether a single card pulled out of the deck will be red or black (50/50 odds). All 52 cards are shuffled after each draw. The last 20 draws have been black. What would the 'vegas' odds be that the next card would be black? I think the vegas odds would be 50/50. In other words, the likelihood that the streak continues is 50/50. No ?

I agree, it's a head scratcher. After 20 black cards in a row, you think there is 'pressure' from somewhere to make a red card show up, lol.
Click to expand...
Well according to AI and I think probability, the odds for each hand (Or card choice -as you mentioned) remain a constant. In other words the cards don't keep track of anything before (So far so good). However, the winning streak is different in that the longer it continues, the probability reduces of it continuing. So after a streak of 50, it is much more likely to fail than a streak of only two in a row.

The problem for me is being able to reconcile those two truths. It seems that after 50 hands, or choices, the odds shouldn't be 50/50 if the streak is more likely to fail than keep going. IDK, I think I just have a mental block there.
 
That Lady in PA said:
https://www.coconino.edu/resources/...and-sciences/MAT142/Chapter_3_Probability.pdf

Gambler's fallacy - Wikipedia
Click to expand...
Thank you for that, and I understand that it is the gambler's fallacy and what it entails, which is what I originally posted. I understand the basic fundamentals, but the difficulty remains. I will use the example from the gambler's fallacy.

The odds of being heads or tails from flipping a fair coin is 50/50. Even the probability of having it come up the same 3 times in a row isn't that unusual. However, having it come up the same after 21 flips is 1 in 2,097,152. so the odds of that happening is extremely remote. Having a roulette wheel come up the same color 26 times in a row has only happened once in history, and the odds of it happening were 1 in 67 million. So, it can occur, but it is astronomically unlikely.

That being the case, it seems that the odds of it not happening on the next spin would be a safer bet than betting it will happen again. Even though the odds still say 50/50, it just doesn't seem to jive with the likelihood of long streaks.
 
Not sure if this helps, but I asked AI the following question:

How can this be true: Out of a hundred hands of blackjack the average would be the dealer wins about 48, and the player wins about 44, and about 8 pushes. The likelihood of the dealer winning all 100 hands (Given normal betting practices) is next to zero. How can the dealers average remain 48% even as the dealer's streak continues?

AI's answer:

The key is that the “48 dealer wins, 44 player wins, 8 pushes” is a long-run average over many independent hands, not a rule that balances itself hand-by-hand.

A streak doesn’t “force” the odds to change on the next hand.

Here’s the intuition:
  • Suppose the dealer has already won 10 hands in a row.
  • The 11th hand is still approximately:
    • 48% dealer win
    • 44% player win
    • 8% push
The cards have no memory of the streak.

This is the same idea as coin flips:
  • A fair coin can land heads 10 times in a row.
  • The next flip is still 50/50.

People often expect the universe to “correct itself” after a streak, but that’s the classic gambler’s fallacy. The probabilities don’t rebalance short-term outcomes.

The reason “dealer wins all 100 hands” is astronomically unlikely is simply because multiplying probabilities repeatedly makes tiny numbers:

0.481000.48^{100}0.48100
0.481000.48^{100}0.48100

That number is incredibly small. But there are many possible sequences near the average:
  • 48 dealer wins / 44 player wins / 8 pushes
  • 52 / 40 / 8
  • 45 / 47 / 8
  • etc.
Those “mixed” outcomes collectively account for almost all probability.

Another important point: “average” does not mean “smooth.”

Even if the true dealer edge is around 48% wins:
  • You can still see runs of 8, 10, or 15 dealer wins.
  • You can also see player streaks.
  • Short-term results fluctuate wildly.
Over a very large number of hands, the proportions tend to stabilize near the expected values because of the Law of Large Numbers:

Observed frequency→True probability\text{Observed frequency} \to \text{True probability}Observed frequency→True probability
Observed frequency→True probability\text{Observed frequency} \to \text{True probability}Observed frequency→True probability

But convergence is slow and noisy. A casino can have long dealer streaks and still end up near 48% dealer wins after hundreds of thousands of hands.

One more subtlety: in blackjack, “dealer wins 48% of hands” does not mean the casino only barely profits. The house edge comes from:
  • players acting first (they can bust before dealer plays),
  • blackjack payout structure,
  • doubles/splits,
  • insurance side bets,
  • rule variations.
So the casino can maintain a small but steady mathematical edge even while individual sessions look chaotic.
 
Gardenlover said:
Not sure if this helps, but I asked AI the following question:

How can this be true: Out of a hundred hands of blackjack the average would be the dealer wins about 48, and the player wins about 44, and about 8 pushes. The likelihood of the dealer winning all 100 hands (Given normal betting practices) is next to zero. How can the dealers average remain 48% even as the dealer's streak continues?

AI's answer:

The key is that the “48 dealer wins, 44 player wins, 8 pushes” is a long-run average over many independent hands, not a rule that balances itself hand-by-hand.

A streak doesn’t “force” the odds to change on the next hand.

Here’s the intuition:
  • Suppose the dealer has already won 10 hands in a row.
  • The 11th hand is still approximately:
    • 48% dealer win
    • 44% player win
    • 8% push
The cards have no memory of the streak.

This is the same idea as coin flips:
  • A fair coin can land heads 10 times in a row.
  • The next flip is still 50/50.

People often expect the universe to “correct itself” after a streak, but that’s the classic gambler’s fallacy. The probabilities don’t rebalance short-term outcomes.

The reason “dealer wins all 100 hands” is astronomically unlikely is simply because multiplying probabilities repeatedly makes tiny numbers:

0.481000.48^{100}0.48100
0.481000.48^{100}0.48100

That number is incredibly small. But there are many possible sequences near the average:
  • 48 dealer wins / 44 player wins / 8 pushes
  • 52 / 40 / 8
  • 45 / 47 / 8
  • etc.
Those “mixed” outcomes collectively account for almost all probability.

Another important point: “average” does not mean “smooth.”

Even if the true dealer edge is around 48% wins:
  • You can still see runs of 8, 10, or 15 dealer wins.
  • You can also see player streaks.
  • Short-term results fluctuate wildly.
Over a very large number of hands, the proportions tend to stabilize near the expected values because of the Law of Large Numbers:

Observed frequency→True probability\text{Observed frequency} \to \text{True probability}Observed frequency→True probability
Observed frequency→True probability\text{Observed frequency} \to \text{True probability}Observed frequency→True probability

But convergence is slow and noisy. A casino can have long dealer streaks and still end up near 48% dealer wins after hundreds of thousands of hands.

One more subtlety: in blackjack, “dealer wins 48% of hands” does not mean the casino only barely profits. The house edge comes from:
  • players acting first (they can bust before dealer plays),
  • blackjack payout structure,
  • doubles/splits,
  • insurance side bets,
  • rule variations.
So the casino can maintain a small but steady mathematical edge even while individual sessions look chaotic.
Click to expand...
Thank you. I got the same info from AI when I originally posted, and I'm not questioning that it's true. It's just that it doesn't make sense to me when I consider both together. If the roulette wheel has come up the same (Red) 26 times in a row only once in all the history of gambling, then the odds of it coming up one more time in red just seems inconceivable and staggeringly unlikely since it's never happened, and yet the odds are still 50/50 that it will. They seem highly contradictory, and yet they are not. I guess it's just one of those things you don't need to understand .... just accept.
 
If it makes you feel any better, here's one that bugs me. If you're moving from point A to point B you first get half way there, then 3/4 of the way there, then 7/8, etc. (You have to keep going half of the remaining distance ) But you never get 'exactly' to point B. Before it or after it, but never exactly there. 🤪 Concepts : infinity and exactness. Does infinity exist? Is there exactness? seems like 1.0 should be exactly 1.0, not approximately 1.0. The concept of 'half' ? Can it be exact? Hmmmm.
 
Last edited:
RandomName said:
If it makes you feel any better, here's one that bugs me. If you're moving from point A to point B you first get half way there, then 3/4 of the way there, then 7/8, etc. (You have to keep going half of the remaining distance ) But you never get 'exactly' to point B. Before it or after it, but never exactly there. 🤪 Concepts : infinity and exactness. Does infinity exist? Is there exactness? seems like 1.0 should be exactly 1.0, not approximately 1.0. The concept of 'half' ? Can it be exact? Hmmmm.
Click to expand...
Yes, that's a head scratcher, because how could you ever go past point B if you never arrived at it? It seems sometimes the math doesn't work, kinda like trying to unite quantum physics with the standard physics model.

Your illustration reminded me of a riddle my dad once told me as a kid.

So these three guys were on a road trip and needed to get rooms for the night, so they stop at a motel.
They ask the owner how much for 3 rooms, and the innkeeper says that will be $30 for all, so they pool their money and pay him.
Later the innkeeper feels a bit guilty that he may have overcharged them, so he gives the bellboy $5 to give back to them.

On the way to the rooms, the bellboy realizes that he can't equally divide the $5 between 3 guys, so he keeps $2 for himself and gives each of them a dollar back. So the guys were out $9 apiece for the rooms (9 x 3 = 27, and the bellboy has $2. So what happened to the other dollar?
 
I guess the example is about one on one dealer vs. single player. To keep AI busy what does AI say about the dealer or house odds when the dealer is dealing to 6 players? Or the variable of cards in a shoe. For example one deck favors the player 8 decks favor the casino.
 
Knight said:
I guess the example is about one on one dealer vs. single player. To keep AI busy what does AI say about the dealer or house odds when the dealer is dealing to 6 players? Or the variable of cards in a shoe. For example one deck favors the player 8 decks favor the casino.
Click to expand...
Well, card counting aside, the odds always favor the house no matter how many decks because the dealer has the advantage that others may go bust first. It's only a slight advantage, but it's all they need. I think they only use multiple decks to deter card counters.
 
You must log in or register to reply here.
Back
Top