Total probability and Bayes by code

You will write three functions: one for the law of total probability, one for Bayes' theorem and one for the chance of being ill after a positive test. The checker uses the three machines of the lessons and a screening test.

The idea. For a partition with priors $p_i$ and likelihoods $l_i$, the total probability of $A$ is $\sum p_i l_i$. The posterior of case $k$ is $p_k l_k$ divided by that total. For a test, the two cases are ill and healthy.

Your tasks.

1. Finish total_probability(priors, likelihoods). Add the product of each prior and its likelihood.
2. Finish posterior(priors, likelihoods, k). Return the posterior of case number k, counting from $0$.
3. Finish predictive_value(prevalence, sensitivity, specificity). Return the chance of being ill after a positive test.