Probabilistic Sequence Function

[cryptist]

Let's assume we have a coin. When it is tossed, in first 2 times it comes head, and the next time tails. It goes like that in sequence, let's say two times. 2 head, 1 tails, 2 head 1 tails.. Btw, the coin is not fake, so head and tails both have equal probability of %50.

Is there a function representation of that in mathematics? For example if we say 1 head, 1 tails and goes like that; we may write (-1)^n. (Therefore 1 stands for head and -1 stands for tails) Can we write an analytic function that represents this sequence? (It should also include the probability information)

 

[jbriggs444]

Let's assume we have a coin. When it is tossed, in first 2 times it comes head, and the next time tails. It goes like that in sequence, let's say two times. 2 head, 1 tails, 2 head 1 tails.. Btw, the coin is not fake, so head and tails both have equal probability of %50.

Is there a function representation of that in mathematics? For example if we say 1 head, 1 tails and goes like that; we may write (-1)^n. (Therefore 1 stands for head and -1 stands for tails) Can we write an analytic function that represents this sequence? (It should also include the probability information)

It's a periodic function with three known values. Fitting it to the sin function and rescaling to fit your given encoding gives:

1/3 + 4 sin ( 2 pi (x/3+1/12) ) / 3

For x = 0 that's 1/3 + 4 sin ( pi/6 ) / 3 = 1/3 + 2/3 = 1
For x = 1 that's 1/3 + 4 sin ( 5pi/6 ) / 3 = 1/3 + 2/3 = 1
For x = 2 that's 1/3 + 4 sin ( 3pi/2 ) / 3 = 1/3 + -4/3 = -1

It's not clear what this has to do with probability. It's deterministic.

 

[cryptist]

It's a periodic function with three known values. Fitting it to the sin function and rescaling to fit your given encoding gives:

1/3 + 4 sin ( 2 pi (x/3+1/12) ) / 3

For x = 0 that's 1/3 + 4 sin ( pi/6 ) / 3 = 1/3 + 2/3 = 1
For x = 1 that's 1/3 + 4 sin ( 5pi/6 ) / 3 = 1/3 + 2/3 = 1
For x = 2 that's 1/3 + 4 sin ( 3pi/2 ) / 3 = 1/3 + -4/3 = -1

It's not clear what this has to do with probability. It's deterministic.

Great! Yes, it is deterministic actually. The probability comes here: This function has two possible values; 1 and -1, with probabilities 2/3 and 1/3 respectively. Let's apply an operation to this function so that, it shows us the probabilities of its values. Is there such operation that leads us to probabilities of the values of that function?

 

[jbriggs444]

The asymptotic density of x values where this function evaluates to 1 is 2/3.
The asymptotic density of x values where this function evaluates to -1 is 1/3.

The asymptotic density of a subset of the natural numbers is the limit (if it exists) of the number of elements in the subset that are less than n taken as a fraction of n as n increases without bound.

The notion of asymptotic density is not the same thing as "probability", though there are similarities.

 

[CellCoree]

I can provide a response to the concept of a probabilistic sequence function. In this scenario, we are considering a coin that is tossed multiple times and the resulting sequence is 2 heads followed by 1 tail, and then repeating. The question is whether there is an analytic function that can represent this sequence, taking into account the probability information.

Firstly, it is important to note that a sequence is a specific order or pattern of events, and a function is a mathematical relationship between inputs and outputs. In this case, the sequence is determined by the outcomes of the coin toss, while the function would represent the relationship between the number of tosses and the resulting sequence.

In mathematics, there are various ways to represent a sequence, such as using a recursive formula or a closed-form formula. However, it is not possible to write an analytic function that represents this particular sequence, as it is probabilistic in nature. This means that the outcome of each toss is determined by chance, and cannot be predicted with certainty.

Furthermore, the probability of getting heads or tails on each toss is independent of the previous tosses. This means that there is no underlying pattern or relationship that can be described by a function. Each toss has an equal probability of 50% for heads and 50% for tails, and this information cannot be incorporated into a function.

In conclusion, while we can describe the sequence of 2 heads followed by 1 tail as (-1)^n, there is no analytic function that can represent this probabilistic sequence. The nature of the coin toss and its equal probability for heads and tails make it impossible to predict or describe the sequence using a mathematical function.