Oh, that’s not the only reason. I just got sick of blackjack one day and didn’t feel like shooting craps, so I thought I’d try three-card poker. I know that it’s not the good bet blackjack is, but it’s not a bad choice for a change of pace.
Excellent discussion - this is some first-class fighting of ignorance.
I think this question is related to another concept I have trouble with: the way that probability “evaporates” after the event has happened.
Like this: What’s the chance of getting heads with a single coin toss? Easy, 50%. How about, after a coin comes up heads 3 times, what’s the chance it will turn up heads on the 4th time? Still 50%. HOWEVER, before the coin is tossed: What’s the chance that I toss this coin 4 times, and get 4 heads? (1/2)^4 = 6.25%. I know the stuff about independent events, and the coin not having a memory, but I just have trouble getting my mind around this “viscerally”.
(Even stranger, to me: after the coin toss, “What is the chance that the coin will turn up heads?” 100% (if it did) - where did the probability “go” ?)
I think this is something fundamental about probability that I’m not grasping, even though I know the formulas and can apply them.
If you have trouble with that, get a load of this. We talk about the probability of getting a certain card next, then after it’s dealt, it’s a certainty one way or the other. But that card was already what is was, when it was sitting unobserved on the top of the deck. Even at that point, there was no “probability” in discussing that single event.
So when we talk about probabilities, we’re talking about what would happen after a very large number of independent trials, all averaged together. The law of averages as it’s known. You act in a single event based on probabilities, but it’s kind of strange to talk of the probability of a specific lady’s kids being both male, when those kids are already born.
My favorite example is in cosmology. “What’s the probability of a certain constant being 0.99999999? Nah, it must be 1 exactly, and we should build theories around that.” But what does it mean for anything about the universe as a whole to have probability, when it’s all there is? You can’t repeat and measure observations about the creation of the universe!
Darren, I could ramble on for a while about math and concepts here, but it probably wouldn’t help. You seem to understand all the math and how to apply it, it just doesn’t connect with you intuitively. That’s OK. A lot of statistics isn’t intuitive to a lot of people. I don’t know any way to make it intuitive except to work with it a lot – dream up a theory, and then flip a lot of coins, deal a lot of cards, roll a lot of dice, and observe what actually happens. My guess is that it will start feeling visceral after a couple hundred trials.
The way I think of probability is, it’s a system for making informed guesses. The more information you have, the less you’re required to guess about. Because we humans can’t see into the future, often we gain more information as time moves forward.
So, prior to my flipping a coin, I don’t know whether it’ll come out heads or tails: I lack information, and so I resort to probability theory to make a guess as to the flip’s outcome. After I’ve flipped it, I have that information, and so I don’t need to make a guess.
The probability of the coin’s flip didn’t actually change: whether it was going to come up heads or tails depends on the coin’s weight, on the strength with which I flip it, on the spin I put on the coin, etc., and all those factors in turn depend on other things (who taught me to flip coins, whether my hands are dry, etc.) Since we don’t know any of those factors ahead of time, probability is the way we gain our best guess. But once the coin has flipped, we have a much better means of guessing: instead of looking at probability theory, we look at the coin.
When you flip four coins, before you flip them, you don’t know the outcome of any of them, so again, prob theory gives you a way to make a guess. Once you’ve flipped three of them, you can make your guess by looking at how those three came up, and using prob theory to make a guess about the fourth one. Even though it looks as if the probability changed, it didn’t: all that changed was the information you had available to you when you made your guess.
Does that make sense?
Daniel
LHOD, muttrox - thanks, a lot of that rings very true and gives me food for thought. Especially helpful to me is seeing probability as the measure of what would happen over a large number of trials, on average. I wish someone had explained it intuitively when I was learning the formulas and how to apply them at school.
I remember seeing an analysis once where, if you toss a coin some large number of times, you’re very very unlikely to get exactly 50% of each outcome, but the more trials, the closer you get to it. That is weird to me too. Perhaps it’s our bias towards “round” numbers coming out.
With the large # of coins – the counter-intuitive results is:
The **average ** tends to get closer to 50%
The **absolute difference ** between heads and tails tends to diverge even further.
Illustrative example:
1000 coins - 540 heads, 460 tails = 46% heads, diff of 80
100,000 coins - 502,055 heads, 497,945 tails = 50.2% heads, diff of ~4,000
100,000,000 coins - 500,080,000, 499,920,000 = 50.008% heads, diff of 160,000
502,055 + 497,945 = 1,000,000
500,080,000 + 499,920,000 = 1,000,000,000
Amazing the things you can do with statistics!
If you’re counting cards at a blackjack table, you do have slightly better odds sitting at the “anchor” or “3rd base” position, as you get to see more cards before making your play.
As for poor playing decisions by other players at the table affecting your chances of winning - as has been pointed out, that’s wrong.
Interestingly, the more decks of cards sitting in the shoe, the less chance you have of getting dealt a blackjack.
Strange, but true.
How’s that?
Seeing more cards doesn’t affect your odds. It gives you more information on the distribution of the cards left, so if you’re that rarest of people, a truly knowledgable card counter, you can adjust your play under certain distributions.
I’d also like to see a cite of your statement:
Not a big effect, but true. To illustrate: Suppose your first card is an ace, and you’re playing with one deck. There are now 51 cards left in the deck, of which 16 are “tens” which will complete your blackjack. So you have a 31.37% chance of blackjack. But now suppose that there are ten decks, and your first card is an ace. Now, there are 160 “tens” left, out of 519 cards, or 30.83%. The difference is because, with more decks, you have a slightly higher chance of getting two aces or two tens. But I rather doubt this difference is large enough to justify any modification to optimal strategy.
Cool. Never thought of that before.
With one deck, the probability of getting a blackjack on any hand is 4.83%. With six decks, it’s 4.75%.
Math:
one deck:
p = 4/52 * 16/51 * 2
six decks:
p = 24/312 * 96/311 * 2
double down. 
Oddly enough, this situation just happened to me this weekend. I was at 3rd base, and held a soft 18 against the dealer’s 6. I doubled down, got a lousy (high) card, which “would’ve been the dealer’s bust card”. Got a tsk-tsk from the dealer and filthy looks from the other players at the table.
Knowing that I made the right play didn’t help much.