Sum of two Binomial Distributions

Questions
3

Thanks for the prompt response. I was using one data point just to test if the code was working.
Here are some simulated data which should make the probabilistic model clearer.

fn = .2
fp = .05
n = [100] * 200
contaminated  = np.random.binomial(100,.4)
not_contaminated = 100 - contaminated
true_positive = np.random.binomial(contaminated, (1-fn)) 
false_positive =np.random.binomial(not_contaminated,fp)
y = true_positive + false_positive

with Model() as food_model:
    positives = pm.DiscreteUniform("positives", lower = 0, upper = n)
    proportion_positives = pm.Deterministic("proportion_positives", positives/100.0)
    true_positives = pm.Binomial("true_positives", n = positives, p = (1 - fn))
    false_positives = pm.Binomial("false_positives", n = y - positives, p = fp)
    model = pm.Deterministic('model', true_positives + false_positives, observed = y)
    


with food_model:

    # Draw samples
    trace = sample(1000, njobs=2)
    # Plot parameter
    forestplot(trace, varnames=['proportion_positives'])

print pm.stats.hpd(trace['proportion_positives'], .05)