Probability that x-axis perfectly separates clusters Suppose you have two clusters of points in the 2-D coordinate plane sampled from two Normal distributions. The clusters are centered at (0,1) and (0,-1). Is the probability that the x-axis perfectly separates the two clusters maximized when:
-the # of sampled points is small and the variances of the Normal Distributions are small
-the # of sampled points is large and the variances of the Normal Distributions are small
-the # of sampled points is small and the variances of the Normal Distributions are large
-the # of sampled points is large and the variances of the Normal Distributions are large
I think that it should be the 1st option, because less variance ensures that the points will be closer to each other and the less points sampled, the less probability that one of them will cross the x-axis. Is my thinking correct?

Yes, you are exactly correct.
Here is an illustration, which also shows you can easily simulate questions like this one to sharpen your intuition:

R code:
library(MASS)
set.seed(1)
par(mfrow=c(2,2),las=1)
#
plot(rbind(mvrnorm(10,c(0,1),0.2*diag(2)),mvrnorm(10,c(0,-1),0.2*diag(2))),
    pch=c(rep(19,10),rep(21,10)),xlab="",ylab="",
    main="Small number, low variance")
abline(h=0)
#
plot(rbind(mvrnorm(50,c(0,1),0.2*diag(2)),mvrnorm(50,c(0,-1),0.2*diag(2))),
    pch=c(rep(19,50),rep(21,50)),xlab="",ylab="",
    main="Large number, low variance")
abline(h=0)
#
plot(rbind(mvrnorm(10,c(0,1),0.5*diag(2)),mvrnorm(10,c(0,-1),0.5*diag(2))),
    pch=c(rep(19,10),rep(21,10)),xlab="",ylab="",
    main="Small number, high variance")
abline(h=0)
#
plot(rbind(mvrnorm(50,c(0,1),0.5*diag(2)),mvrnorm(50,c(0,-1),0.5*diag(2))),
    pch=c(rep(19,50),rep(21,50)),xlab="",ylab="",
    main="Large number, high variance")
abline(h=0)

