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.