Hi everyone,

I am trying to model a french vocabulary test, finding the probability of passing, based on a number of factors.
Currently, I am looking at Binomial Distribution and Poisson Binomial Distribution of Bernoulli trials.
The way I am separating my data is based on connection to the English language and word length. My data is coming from previous tests I have done. What I am trying to do is combine the passing probability of different sets of types of words, to find an overall probability.
For Example,
Word length of 5, unrelated-to-English Pass/fail
belle pass
chaid fail
santé pass
avoir pass
and combining this with another set of words.
I think that poisson binomial distribution is the more likely one but I dont know how to apply the formula:
The probability of having k successful trials out of a total of n can be written as the sum [1]
{\displaystyle \Pr(K=k)=\sum \limits _{A\in F_{k}}\prod \limits _{i\in A}p_{i}\prod \limits _{j\in A^{c}}(1-p_{j})}where {\displaystyle F_{k}} is the set of all subsets of k integers that can be selected from {1,2,3,...,n}. For example, if n = 3, then {\displaystyle F_{2}=\left\{\{1,2\},\{1,3\},\{2,3\}\right\}}. {\displaystyle A^{c}} is the complement of {\displaystyle A}, i.e. {\displaystyle A^{c}=\{1,2,3,\dots ,n\}\setminus A}.