1. 5% of samples that are tested for a particular disease return a positive result. To save time (and money) the pathologists pool the samples in batches of 10 and test the whole batch. If the batch returns a negative result, they do not need to test each of the 10 samples one by one. But if the batch returns a positive result, it means that at least one sample in the batch is positive, and all 10 samples need to be tested.

Let X be the number of positive samples in a batch of 10, and Y be the number of batches that are tested until the first postive result.

(a) Name the probability distribution of X and specify its parameter(s).

(b) Calculate the probability that a batch returns a positive result.

(c) Name the probability distribution of Y and specify its parameter(s).

(d) Calculate the probability that at least 4 batches are tested before one returns a positive result.

(e) Let Z be the number of batches tested until 4 batches return a positive result. Name the probability distribution of Z and specify its parameter(s).

(f) Calculate the probability that exactly 15 batches are tested until 4 batches return a positive result.