How to calculate probability of these two pair-of-sums ($S_{n}$ and $T_{n}$) and ($SEvenF_{n}$ and $SOddF_{n}$) being the same? Say we have a sequence of $n$ positive integers, we can assume they're randomly chosen, let's call it $U_{n}$.

Let $S_{n}$ = sum of $U_{n}$ from $1$ to $n$.

Let $T_{n}$ = sum of $n$ from $1$ to $n$. In other words, $T_{n}$ is the triangle number $T_{n} = \frac{1}{2}n(n+1)$.

Define $F_{n} = U_{n} + n$.

(note that $SF_{n}$, sum of $F_{n}$, is equal to $S_{n} + T_{n}$)

Define two sets $EvenSet$ and $OddSet$ containing $F_{n}$ when it is even and odd respectively.

Let $SEvenF_{n}$ be the sum of numbers in $EvenSet$ and $SOddF_{n}$ be the sum of numbers in $OddSet$.

(note that $SEvenF_{n} + SOddF_{n} = SF_{n} = S_{n} + T_{n}$)

What is the probability of the two pairs ($S_{n}$ and $T_{n}$) and ($SEvenF_{n}$ and $SOddF_{n}$) are the same pairs?

In other words, either "$S_{n} = SEvenF_{n}$ thus $T_{n} = SOddF_{n}$" or "$S_{n} = SOddF_{n}$ thus $T_{n} = SEvenF_{n}$".

How to calculate this probability?

PS: I have a sample data with $n = 114$ in here
http://goo.gl/k96FZ

PS2: Is there a way or an algorithm to generate such sequence?

PS3: @ZefChonoles has correctly pointed a concern about Probability Measure. Originally I intended this question to have no restriction on the positive integer set, that is to work on the infinite set of $\mathbb Z^{+}$. 

But I believe you are free to impose certain restriction to this problem, as long as you can share insights on approaches in solving this problem. 

For example, you can restrict $U_{n}$ and $F_{n}$ to be within $[1, 400]$ for the $n=114$ case. Or you can restrict them to be within $[1, Cn]$ with some constant $C$ for the general case. Or within $[1, Tn]$. Anything that helps, basically. Thanks!
The integer wont exceed 3300
If Sn = S even Fn;
Sn = S even n + S odd n
S even F n = S even n + T even n

Then 
S even n + S odd n = S even n + T even n
S odd n = T even n
T even n is 3306
S odd n = 3306
The sum of 57 random positive integer is 3306.
Is it safe to use combination of 3306 and 57? 
As for the other one:
Tn = S even Fn
Tn = T even n + T odd n
S even Fn = S even n + T even n
Thus
T even n + T odd n = S even n + T even n
T odd n = S even n
T even n is 3249
S even n = 3249

Combination of 3249 and 57

What I still wonder is, how to determine all possibility used in this to divide them
The integer wont exceed 3300
If Sn = S even Fn;
Sn = S even n + S odd n
S even F n = S even n + T even n

Then 
S even n + S odd n = S even n + T even n
S odd n = T even n
T even n is 3306
S odd n = 3306
The sum of 57 random positive integer is 3306.
Is it safe to use combination of 3306 and 57? 
As for the other one:
Tn = S even Fn
Tn = T even n + T odd n
S even Fn = S even n + T even n
Thus
T even n + T odd n = S even n + T even n
T odd n = S even n
T even n is 3249
S even n = 3249

Combination of 3249 and 57

What I still wonder is, how to determine all possibility used in this to divide them