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Learning she was positive, Alice notified the authorities and reported her case and the fact that she had five close contacts. The authorities proceeded to trace them and test them. Bob is one of Alice's close contacts, but unlike Alice, Bob doesn't have Covid. Bob still has to be tested for Covid, however. Bob will undergo the same testing procedures as Alice will, because he's likely to have Covid.

Before we are able to discuss probability problems for the non-negative people, however, we need to introduce two more elementary events and their probabilities.

Let C be the event that a Covid-negative person will get a negative in the "preliminary test", and let D be the event that a Covid-negative person will get a negative in the "confirmatory test". This is necessary because one isn't able to find out these probabilities based on A and B alone, which are probabilities for the Covid-positive.

Suppose P(C) = 0.95, and P(D) = 0.99.

1. Express the following in terms of events C and D, and then find the probability of occurrence of each of these events:

   1. The event that Bob will be determined negative under the usual testing procedures

   2. The event that Bob will be determined positive under the usual testing procedures (a false positive)

Suppose restrictions are relaxed, so confirmation is done by testing again with the first test. Then in the following two items, P(D) = 0.95.

3. The event that Bob will be determined negative under the relaxed testing procedures

4. The event that Bob will be determined positive under the relaxed testing procedures (a false positive)

2. Suppose that the cost of the cheaper test is five hundred pesos, while the gold standard test costs three thousand pesos. Under the usual testing procedures, one will end up either taking only the first test (cheaper one) if it results positive, or both tests should the first test end up negative. Under relaxed testing procedures, they can take the cheaper one twice. Find the expected cost for testing Bob under

   1. the usual testing procedures.

   2. the relaxed testing procedures.

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