Fractional Dice - Investigating Discontinuity in Nature

[DaveC426913]

I hate discontinuity in nature.

When I graph the probabilities of dice rolls (say, 2 dice or 3 dice) I get integer values that result discrete steps, like a staircase.

But I know that the set of points represents a smooth bell curve. How can a set of discrete integer points - each of which is separated by a gaping chasm of an infinite set of fractional numbers - impersonate a smooth continuum?

So I keep finding myself asking: is there meaning to the fractional points along the bell curve between the integers? Is there a such thing - at least in principle - as fractional results of dice roills?

 

[Werg22]

A dice of roll does not engender data that are fit for a bell curve, since every result has just as much chances of occurring as the other. Anyway, a set of events (numbers) that follows a normal distribution hypothesis will have to contain allot of events and in different frequencies. This said, a normal distribution is part of mathematical modeling and is basically a device that allows us to work with a continuous function rather than a discrete one. There is no ambiguity: the normal distribution "extends" the range of probability: instead of having a finite number of probabilities because of a finite number of elements in set, we now have an infinite number of probabilities because the set's argument has been made a continuum.

 

[DaveC426913]

A dice of roll does not engender data that are fit for a bell curve, since every result has just as much chances of occurring as the other.

I never said one die. I said two or three. (I guess what I didn't say was 'the sum of')'

This said, a normal distribution is part of mathematical modeling and is basically a device that allows us to work with a continuous function rather than a discrete one. There is no ambiguity: the normal distribution "extends" the range of probability: instead of having a finite number of probabilities because of a finite number of elements in set, we now have an infinite number of probabilities because the set's argument has been made a continuum.

I confess I am not sure if this answers my question or not.

 

[Werg22]

That's what I meant too. If you roll a dice twice, you have just as much chances of getting 2 as to getting 12.

The things is: the so called values between the integers are given a meaning. Once you consider a normal distribution, the inital set of values does not mean anything, and hence everything that is "meaningful" is given by the curve, not the set anymore. In other words, the non-integers become just as meaningful as the integers.

 

[robert Ihnot]

Werg22: That's what I meant too. If you roll a dice twice, you have just as much chances of getting 2 as to getting 12.

You have much more chance of getting a 7. (However, these matters are very erratic in actual practice for just a few tosses.) Coin tosses, heads and tails have been frequently studied with a Bell curve. It's just a question of realizing that the binominal results for coin flips can be turned into probability density function as numbers increase.

For instance, Warren Weaver in "Lady Luck," extensively goes over the matter of the number of heads as the number of tosses increases. In 10,000 tosses, we would expect around 10000!/(5000!)^2 cases of 5000 heads. Dividing this by 2^10,000 gives a probability density function as we go over a unit length. "By making enough trials we can essentially secure that the ratio of successes to total trials will closely approximate the probability...and by "normalizing" the curve, using the standard deviation to shrink the horizontal and stretch the vertical, it becomes more and more like the bell shaped curve."

 

[AlephZero]

If you have a continuous variable, it makes no practical sense to talk about the probability of one particular value - e.g. the probability that the length of an object is 1.845842548275067434324 meters (exactly!) is zero. The meaningful practial idea is the probabilty that the length is in a given range.

If you have a discrete distribution like the probabilities or 0, 1, 2, ... 10,000 heads from 10,000 tosses (Binomial, p = 0.5, n = 10,000) and a continuous distribution (Normal, mean 5,000, standard deviation 50) it makes sense to say "the discrete probability of getting 5678 heads is approximately the same as the Normal probability that the continuous variable is between 5677.5 and 5678.5".

That's thinking about it the opposite way round to your "fractional results of dice rolls" (which doesn't make much sense to me).

Also, when you plot "probability distributions" of discrete and continuous variables you are usually plotting two different things. In the discrete case you can plot the probability of each separate event as a bar chart. In the continuous case, the corresponding plot is the probability density function. To turn that into a probabilty, you have to integrate over a range. The two plots look similar shapes but they are plots of two different things.

For the scores from 2 or 3 dice, the normal distribution is a poor approximation to the discrete probabilities however you might try to use it, but that's another issue - google "Central Limit Theorem" for more about that.

 

[robert Ihnot]

AlephaZero:you have to integrate over a range. The two plots look similar shapes but they are plots of two different things.

I went ahead and changed a few things I had written above to try and take that into account, changing from a discrete situation to a probability density function over a unit area.