100x odds: pass only vs pass and come

Quote: cowboy

There's still free buffets??

comped ones.

Gambler's Fallacy is the mistaken belief that prevalence or absence of specific outcomes in past events will influence prevalence or absence in future events. Causation across time is disallowed by the notion of independent events. Clustering of outcomes is entirely distinct from Gambler's Fallacy. Clusters can be mathematically described over infinite future events. However, the dispersal of those clusters across time is random and cannot be predicted using mathematics.

In craps for example, the magnitudes and frequencies of futures clusters of Sevens can be described but the timing of those clusters cannot be described. Over the infinite future of dice tosses, we can describe the proportion of all Sevens which will occur with zero, one, two, or any other number of intervening non-Seven tosses. Again, the timing of those clusters is logically unknowable.

Although the future may be opaque to those who limit themselves to logic, a recent spate of good luck may yet serve as a spur to boldness. In poker, a Rush of good hands can be more than mere historical oddity. In craps, perhaps a blizzard of Sevens may disincline one toward Place bets.

Sounds exactly like an eloquent version of Gambler’s fallacy.

Let’s see an example backed up by math instead of superstition.

Quote: FleaStiff

But surely sometime the girl will not be from an agency and the dealer will not have 21 when I have 20 and the free buffet will be fresh and tasty and the drinks will not be watered down.

Yes, sir! rofl - you could not have selected better instances of "Gambler's Fallacy".

Ace2 desires to see a mathematical example of exactly what, specifically?

Quote: pwcrabb

Ace2 desires to see a mathematical example of exactly what, specifically?

For instance you state that a recent blizzard of sevens may disincline one to make place bets. Please provide a mathematical explanation of why that may be the case.

Just math please.

From the viewpoint of a bettor making Place bets, let us define a Short Seven as occurring the first, second, third, or fourth toss following the previous Seven. If we use the subjective estimate of (1/6) for the probability of Seven, such undesirable outcomes together represent 51.78 percent of all future Sevens. This result is sufficiently close to 50.00 percent to permit application of theorems of conditional probability using a binary model with uniform prior probabilities.

In his Memoir on the Probability of the Causes of Events [Statistical Science 1 (1986) 359-378] in which he expanded on Bayes' results, Laplace (1774) derived the following for a frequentist estimation of future probability: Using equal prior probabilities in a binary model, and given (m) successes in (n) trials, the probability of success on the next trial is (m+1) / (n+2).

This ratio is the deceptively simple result of a series of calculations using combinatorics and integrals over a binomial distribution of arbitrarily large degree. This result is sufficiently robust to permit estimations using non-uniform subjectively derived priors as well.

Note that (m+1) / (n+2) approaches (m/n) as both m and n become large. Note also that if (m) = (n) then this ratio approaches 1.

Place bets become increasingly ill-advised as the number of consecutive Short Sevens grows.

Quote: pwcrabb

In his Memoir on the Probability of the Causes of Events [Statistical Science 1 (1986) 359-378] in which he expanded on Bayes' results, Laplace (1774) derived the following for a frequentist estimation of future probability: Using equal prior probabilities in a binary model, and given (m) successes in (n) trials, the probability of success on the next trial is (m+1) / (n+2).

Here's what Wikipedia says about this:

" Under the assumption that little or nothing is known a priori about the relative plausibilities of the outcomes, Laplace derived a formula for the probability that the next trial will be a success.

Pr(next outcome is success)= (s+1) / (n+2)
where s is the number of previously observed successes and n is the total number of observed trials. It is still used as an estimator for the probability of an event if we know the event space, but have only a small number of samples."

Of course, this doesn't apply here, since we know the probabilities involved. Applying that formula to coin flips, if we had 6 heads in 10 flips, we would calculate the probability of a head in the next roll as (6+1)/ 10+2) = .583, or if 4 in 10, then (4+1) / (10+2) = .417. Absurd.

Quote: pwcrabb

Place bets become increasingly ill-advised as the number of consecutive Short Sevens grows.

Gambler's Fallacy II
Cheers,
Alan Shank

I'm enjoying the language/the writing.

Who cares if the conclusion is right or wrong?

Lol.

pwcrabb, you're with NASA, I'm guessing?

Goatcabin Alan Shank cheerfully suggests that the probabilities for real devices are known and settled a priori by hypothesis. The majority of practicing professional statisticians would disagree. They support the historically dominant view that testable probabilities can be acceptably established only following extensive trials. This view is known as the Frequentist school of thought.

The fairness of neither dice nor coins can be assumed ex ante. Only hypothetical devices can be so specified. Laplace aimed his thesis directly at the tossed hypothetical coin of unknown fairness.

Given Goatcabin's unbalanced results even in his very limited sample space, wise gamblers would adjust their future wagers on tosses of that particular coin. Persistent imbalances become increasingly persuasive.