I am having trouble understanding the formal definition of the binomial distribution. $$f(k;n,p) = \Pr(X = k) = {n\choose k}p^k(1-p)^{n-k}$$
Or rather how I "transform" the definition to suit my needs. I randomly conduct 100 experiments in a Java programm with the binomial distribution. The probability of success is 40%. How would I formally write that the results are stored in a vector as is: $$(1,0,1,0,0,0,...,experiment_{100})$$
Let's assume that the experiments you conduct in your Java program are "independent" and put $X :=$ "number of $1$s in the vector from your question". Then $X$ is a random variables which takes values from the set $\{0,\ldots,100\}$ and is distributed according to the Binomial distribution with parameters $n = 100$ and $p = 0.4,$, i.e. $Prob(X = k) = f(k;100,0.4)$ for $k \in \{0,\ldots,100\}$ with $f(\cdots)$ as in your question.