Let X1 ~ Bernoulli (theta) be the indicator that a tree species occupies a forest and Theta be in the set [0,1] denote the prior occupancy probability. The researcher gathers a sample of n trees from the forest and X2 belongs to the species of interest. The model for the data is X2|X1 ~ Binomial (n, lambda, X1) where lambda is in the set [0,1] the probability of detecting the species given it is present. Give expressions in terms of n, theta, and lambda for the following joint, marginal and conditional probabilities:

a) Prob(X1=X2=0)

b) Prob(X1 = 0).

c) Prob (X2=0).

d) Prob(X1=0 | X2=0).

e) Prob(X2=0 | X1=0).