Probability based on Observed Data I'm trying to figure if my reasoning and application of probability is correct in this made up example using R. Example data:
A = c(34.5, 34.2, 35, 35.1, 36, 35.2, 35.7, 34.8, 34.9, 34.4)
B = c(35, 34.2, 36, 35, 34, 34.2, 34.2, 34.5, 34.4)

Let's assume 2 groups (A and B) we sampled some particular measure of interest. Traditional Frequentist approach would be to apply a t-test, given 2 variables and we want to know the probability of observing a difference larger than the one we got, assuming the null-hypothesis is true, and if we were to sample an infinite # of times.
t.test(A,B)

        Welch Two Sample t-test

data:  A and B
t = 1.3423, df = 16.183, p-value = 0.198
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.2131596  0.9509374
sample estimates:
mean of x mean of y 
 34.98000  34.61111 

Simple, but doesn't say much about my observed data. So where I'm stuck mathematically and philosophically is looking at the probability, given the data, of someone in group A having a larger measured value than someone in group B. Because this is 2 groups, I can't do a pre & post comparison. So I used the outer() function for this comparison:
outer(A,B,">")
       [,1]  [,2]  [,3]  [,4] [,5]  [,6]  [,7]  [,8]  [,9]
 [1,] FALSE  TRUE FALSE FALSE TRUE  TRUE  TRUE FALSE  TRUE
 [2,] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
 [3,] FALSE  TRUE FALSE FALSE TRUE  TRUE  TRUE  TRUE  TRUE
 [4,]  TRUE  TRUE FALSE  TRUE TRUE  TRUE  TRUE  TRUE  TRUE
 [5,]  TRUE  TRUE FALSE  TRUE TRUE  TRUE  TRUE  TRUE  TRUE
 [6,]  TRUE  TRUE FALSE  TRUE TRUE  TRUE  TRUE  TRUE  TRUE
 [7,]  TRUE  TRUE FALSE  TRUE TRUE  TRUE  TRUE  TRUE  TRUE
 [8,] FALSE  TRUE FALSE FALSE TRUE  TRUE  TRUE  TRUE  TRUE
 [9,] FALSE  TRUE FALSE FALSE TRUE  TRUE  TRUE  TRUE  TRUE
[10,] FALSE  TRUE FALSE FALSE TRUE  TRUE  TRUE FALSE FALSE

Basically running every possible comparison, asking if A > B, between the 2 groups. If I apply the mean() to this:
mean(outer(A,B,">"))
[1] 0.6666667

I get a probability 66.67%, that someone in group A will have a measured value higher than someone in group B.
My question, is this mathematically appropriate (doing all comparisons)? and if possible on this SE, philosophically reasonable? Again, I'm only speaking of observables, not "possible" long runs, and only possible interpretation given my collected (observed) data. I know the conclusions could change if I collected more data, especially with such a small sample size. I wanted to know if I was in trouble either mathematically or philosophically speaking.
Edit:
Another way of saying what I wrote above is P(A-B>0). I could also ask the P(A-B>x), where x could be a # of interest or relevant to the field. Example:
mean(outer(A,B,"-")>0.5)
[1] 0.4555556

Where P(A-B>0.5) = 45.56%. Numbers are of course very close, and still up to a user to decide if this is useful for their applications or not.
 A: $t$-test compares means of the two samples, but this doesn't mean that means are the only thing that we can compare. You can define your test statistics in terms of probability that any value of A is greater then any value of B (I am not discussing in here if it is valid in your particular case or not, as this may depend on your particular problem), however this is not a test yet!
If you want to conduct a hypothesis test, you need to check what would be the probability of observing the result that from your sample if it came from the null distribution. In terms of your data, the simple nonparametric test you could conduct is the permutation test, where you would mix your two samples together, shuffle them, and then draw the two samples from the shuffled data. By doing so, you would simulate the situation where each of the observations could have been assigned to any of the two groups, i.e. the group assignment would be "random" and should not affect the result of comparison between the groups. By conducting many such simulations, you would obtain a "null distribution" of the test statistic of your choice calculated on the simulated data. Next, you would compare the test statistic calculated on your actual data to the null distribution to see what would be the probability of observing your result if the group assignment played no role.
Notice however that if you wanted to test the hypothesis that

a randomly selected value from one sample will be less than or greater
  than a randomly selected value from a second sample

there already is the Mann–Whitney $U$ test that does exactly this.
set.seed(123)

A <- c(34.5, 34.2, 35, 35.1, 36, 35.2, 35.7, 34.8, 34.9, 34.4)
B <- c(35, 34.2, 36, 35, 34, 34.2, 34.2, 34.5, 34.4)
AB <- c(A, B)

k <- length(A)
n <- length(AB)

Tstat <- function(x) {
  A <- x[1:k]
  B <- x[(k+1):n]
  mean(outer(A,B,">"))
}

Tstat_sim <- replicate(50000, Tstat(sample(AB)))

mean(Tstat(AB) <= Tstat_sim)
## [1] 0.07672


