Probability help needed I have a R data frame (>2000 obs)
       Date meanLVL.QLD meanLVL.VIC
1 1990-01-02       1.194       0.721
2 1990-01-03       1.192       0.715
3 1990-01-04       1.196       0.708
4 1990-01-05       1.206       0.703
5 1990-01-06       1.200       0.708
6 1990-01-07       1.200       0.701

How can I calculate probability that the meanLVL.QLD is higher than the meanLVL.VIC if randomly selects one of the dates from these dataset?
and the probability that between 200 and 500 (inclusive) of the dates, meanLVL.QLD to be higher than meanLVL.VIC? Any assumptions required?
 A: There are a couple ways to test if one is higher.


*

*assume that both of your variables are normally distributed (you can test this assumption using a normality test), and do a one-tailed paired $t$ test.  This tests for the difference in means, so $\mu_{QIC}>\mu_{VIC}$.
do this using t.test(meanLVL.QLD,meanLVL.VIC,paired=TRUE, alternative = "greater")

*You can do a one-tailed non-parametric test (Wilcoxon signed-rank test), but you will be testing the hypothesis that the mean difference of ranks is higher for QIC than VIC and not exactly that the mean on QIC is higher. However, you do not have to assume a particular distribution for your data so there are fewer assumptions.
The probability that you will draw a date with meanLVL.QLD > meanLVL.VLC from your database is equal to the number of cases where this is true devided by the number $N$ of dates in your dataset. 
As for the last step I'm guessing you want to know the probability that you got between 200 and 500 cases by chance alone (choose $p = .5$), or that you will get that in a future dataset with N cases (choose $p$ calculated  above). You can check the probability that the number $X$ out of $N$ dates of meanLVL.QLD > meanLVL.VIC will will be between 200 and 500 assuming that in the true probability of that happening on any one date is p by using the binomial distribution:
$$ Pr(200\leq X\leq 500) = \sum_{i = 200}^{500} \binom{N}{i}p^{i}(1-p)^{N-i}$$  
you can do this by using the cumulative binomial function in R: pbinom(500, size=N, prob=p) -pbinom(199, size=N, prob=p) just replace N with the number of observations in your dataset, and replace p with the true population probability that you want.
