Dice Next Red

Knowing that the probability that Red will show on the very next dice roll is ( 6/36 ) and that it will NOT show is ( 30/36 ), we can specify the probability that the next Red will show on any particular future roll no matter how distant. For example, the probability that the next Red will show precisely on the third future roll is ( 30/36 )( 30/36 )( 6/36 ). Moreover, we can accumulate those probabilities. For example, the accumulated probability that the next Red will appear on or before the fourth future roll is 0.5177.

The probability Distribution Mean is of course 6.0 future rolls. However, use of the Mean as a descriptive Average is appropriate mostly for roughly symmetric distributions. For highly asymmetric distributions, especially those that are highly skewed, the Distribution Median is the superior descriptive Average. By definition, half of all events will occur before the Median and half will occur after the Median. For the future next Red, the Median is 3.8 future dice rolls.

Can this knowledge play a part in bet selection and especially bet timing? Is it useful for appreciating why Reds seem to appear in clusters?

Quote: pwcrabb

Can this knowledge play a part in bet selection and especially bet timing? Is it useful for appreciating why Reds seem to appear in clusters?
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No, and no. Well, the first "no" is based on the assumption that you are either (a) betting on whether a particular roll is red, or (b) considering Martingaling or some other "keep raising your bet until you win" strategy.

I’ll also go with “no.” My view is each roll is an independent event and each roll has exactly the same probabilities for outcomes. (I know many craps players disagree.) Probabilities don’t accumulate making an event “due.” While reds may seem to appear in clusters, I believe that perception is just a human trait. From the Wikipedia article about apophenia:

Quote:

Apophenia has also come to describe a human propensity to unreasonably seek patterns in random information, such as can occur while gambling.

Assuming you are asking this question seriously. Take the same problem of a coin flip, you can do the same statistics for the exact probability for the 3rd coin to be heads. There is no difference in general logic.

The answer to your question is the Gambler's fallacy.

Quote: Wikipedia

If a fair coin is flipped 21 times, the probability of 21 heads is 1 in 2,097,152. The probability of flipping a head after having already flipped 20 heads in a row is 1/2.

Assuming a fair coin:

The probability of 20 heads, then 1 tail is 0.520 × 0.5 = 0.521
The probability of 20 heads, then 1 head is 0.520 × 0.5 = 0.521
The probability of getting 20 heads then 1 tail, and the probability of getting 20 heads then another head are both 1 in 2,097,152. When flipping a fair coin 21 times, the outcome is equally likely to be 21 heads as 20 heads and then 1 tail. These two outcomes are equally as likely as any of the other combinations that can be obtained from 21 flips of a coin. All of the 21-flip combinations will have probabilities equal to 0.521, or 1 in 2,097,152. Assuming that a change in the probability will occur as a result of the outcome of prior flips is incorrect because every outcome of a 21-flip sequence is as likely as the other outcomes. In accordance with Bayes' theorem, the likely outcome of each flip is the probability of the fair coin, which is 1/2.

Gambler's Fallacy characterizes misperception of unchanging probabilities in light of past results. It accurately describes a human irrational response to past results. If in contrast no results have yet occurred, then the Fallacy does not apply. There are no past results in a probability distribution. The probability distribution of Next Red is entirely prospective. It has no past results. Perhaps confusingly, it does not focus upon any single future roll outcome. It simultaneously considers multiple future rolls jointly as fully independent constituents of a single mathematical object. That single mathematical object is the probabilistic event being evaluated.

Consider a parlay bet on multiple future football games, none of which has yet occurred. Their prospective outcomes may be jointly analyzed as a single mathematical object. Multiple future dice rolls may be analyzed analogously.

Quote: pwcrabb

Knowing that the probability that Red will show on the very next dice roll is ( 6/36 ) and that it will NOT show is ( 30/36 ), we can specify the probability that the next Red will show on any particular future roll no matter how distant. For example, the probability that the next Red will show precisely on the third future roll is ( 30/36 )( 30/36 )( 6/36 ). Moreover, we can accumulate those probabilities. For example, the accumulated probability that the next Red will appear on or before the fourth future roll is 0.5177.

The probability Distribution Mean is of course 6.0 future rolls. However, use of the Mean as a descriptive Average is appropriate mostly for roughly symmetric distributions. For highly asymmetric distributions, especially those that are highly skewed, the Distribution Median is the superior descriptive Average. By definition, half of all events will occur before the Median and half will occur after the Median. For the future next Red, the Median is 3.8 future dice rolls.

Can this knowledge play a part in bet selection and especially bet timing? Is it useful for appreciating why Reds seem to appear in clusters?
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Reds appear in clusters because if they were uniformly spaced, you could be sure the dice were crooked. Statistics can calculate the frequency of clusters of a certain length. You are calculating the frequency of non-Red clusters of a certain length. None of this changes the probability of red on the next roll or the Nth spin looking forward. You cannot use the statistics of clusters to gain an advantage.

After N rolls of non-Red, you only know that there is a probability of 1/6 that the sequence of non-Red ends at N rolls. How can you use this knowledge to your advantage? After all, the same thing was true after N=0.

Clusters of events may be described retrospectively of course as a matter of observation and record. Clusters may also be described prospectively by applying a branch of probability theory called Combinatorics.

Forecasting the number of clusters of a specific size is entirely distinct from calculating the probability of one cluster of that specific size. Whether one is forecasting clusters of Tails outcomes in a coin toss experiment or clusrers of galaxies in a region of space, the techniques of Combinatorics apply.

If we define clusters of Red as two or more Reds separated by either zero or one intervening non-Red, then how many such clusters may we expect in a region of 4000 future dice rolls? This is a solvable problem which requires the use of the Next Red distribution.

Quote: pwcrabb

Clusters of events may be described retrospectively of course as a matter of observation and record. Clusters may also be described prospectively by applying a branch of probability theory called Combinatorics.

Forecasting the number of clusters of a specific size is entirely distinct from calculating the probability of one cluster of that specific size. Whether one is forecasting clusters of Tails outcomes in a coin toss experiment or clusrers of galaxies in a region of space, the techniques of Combinatorics apply.

If we define clusters of Red as two or more Reds separated by either zero or one intervening non-Red, then how many such clusters may we expect in a region of 4000 future dice rolls? This is a solvable problem which requires the use of the Next Red distribution.
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There are other ways of solving this problem without using the next-red distribution. The solution is of no practical importance for gaining an advantage. There are so many ways to gain an advantage over casinos that actually work and also allow you to parade your knowledge of basic statistics. Loss rebates are quite common. Figuring out if they are +EV requires statistics or MC simulations.

Quote: pwcrabb

…Consider a parlay bet on multiple future football games, none of which has yet occurred. Their prospective outcomes may be jointly analyzed as a single mathematical object. Multiple future dice rolls may be analyzed analogously.
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But once the parlay bet is made, it cannot be modified.

There’s no similar bet on dice. Most bets can be made or removed at any time. Or are you thinking about creating a No Red For X Rolls type side bet?

Hmmm… Now that I think about it…

A side bet of no Red for (x) number of rolls would be fun. Go for it, DJ !

In other news, no rational technique can gain an advantage in Craps, and none is offered. The House Advantage remains immutable.

I've gone to the Darkside myself, and playing this way you really notice the clumping of 7 rolls [because darksiders don't want a 7-out followed by a come-out 7] Whether you get that clumping plague or not has a lot to do with whether your session is successful or not, it's a complete turn-around of luck to not get the clumping [often].

Unfortunately, I have to conclude knowing this does not help you at all. The dice don't know what they have just been doing. So the history of recent clumping just does not mean the dice are going to keep it up or not ...

Quote: pwcrabb

A side bet of no Red for (x) number of rolls would be fun. Go for it, DJ !

In other news, no rational technique can gain an advantage in Craps, and none is offered. The House Advantage remains immutable.
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I know how to use a betting sequence to beat Interblock bubble e-craps. However, it results in too many hand pays for my taste. That alone would bring a lot of unwanted attention to the play. The exact EV depends on the betting limits. I believe my method is 'rational'.

Sadly, no amount of information regarding past dice rolls can help us rationally regarding future dice rolls. All that we can do is to note tendencies, choose a strategy, and then hope.

Best Wishes on the Dark Side, OdiiousGambit. May your Reds usually be four rolls apart.

Quote: pwcrabb

Sadly, no amount of information regarding past dice rolls can help us rationally regarding future dice rolls. All that we can do is to note tendencies, choose a strategy, and then hope.

Best Wishes on the Dark Side, OdiiousGambit. May your Reds usually be four rolls apart.
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Thanks. Last time out I went 6x odds on a $25 table, which I'll maybe blog about. I didn't get the clumping and also avoided another irritating thing the dice like to do to mess with a darksider, and that's repeating the same number generally. You can see how you don't want that repeating business no matter what number!

To strictly rational players, a history of recent dice outcomes is as useless as a history of recent roulette outcomes, which many roulette tables nonetheless provide for the pleasure of non-rational players. Only the future really matters. What can be deduced about that future?

Place bets are very popular. Their only enemy is Red. Assuming their uninterrupted Working status, Place bets are likely to survive the very nexf dice roll. They are likely to survive over the interval that includes the next three dice rolls. They are NOT likely to survive over the interval that includes the next five dice rolls.

Quote: pwcrabb

…Place bets are very popular. Their only enemy is Red. Assuming their uninterrupted Working status, Place bets are likely to survive the very nexf dice roll. They are likely to survive over the interval that includes the next three dice rolls. They are NOT likely to survive over the interval that includes the next five dice rolls.
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Yeah, but if they survive those first three rolls, then, once again ignoring the past, they again become likely to survive the next three rolls.

DJ you are entirely correct. All past rolls always become irrelevant. The forecast probably safe interval renews each roll. Red eventually appears despite that repeated renewal.

Many players have some personal signals for when to take down their Place bets. Most players do not.

Quote: pwcrabb

To strictly rational players,

So about one in ten thousand craps players

Dice is always random. I switched to the dark side a few months ago.

Quote: pwcrabb

To strictly rational players, a history of recent dice outcomes is as useless as a history of recent roulette outcomes, which many roulette tables nonetheless provide for the pleasure of non-rational players.

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I can think of at least one person in this forum that would take offense to this....LOL