**1. A cooperation model.**

Consider the following two-species model.

where , and are positive constants.

- Under what circumstances is there a positive coexistence equilibrium?
- In the case where there exists a positive equilibrium, find ALL equilibria, determine their stability and sketch the phase plane.
- What does the phase plane look like when there is no positive coexistence state?

**2. On the half life of one-hit wonders**

Suppose that there is a band that puts out a great first album, but then they soon run out of ideas. The first album has 15 songs, of which 5 are good. The second album has 12 songs, of which 3 are good. The last album has 10 songs, of which only 1 is good. Use Bayes’ Theorem to answer the following questions.

- Suppose that a radio station picks songs at random to play and you hear one of their good songs on the radio. What is the probability that song is from the first album?
- Suppose that another radio station plays only good songs and chooses from them at random. You hear one of the band’s good songs on this station. What is the probability the song is from the first album?

**3. Think about this one carefully!**

You visit a neighbor’s house. You know the neighbor has two children. When you knock, a girl answers the door. What is the probability that the other child is a girl?

**4. Bayes’ Theorem in the news.** Find an example in the news where a test is given widely to many people where there is little to no indication that the person being tested has the condition. Some recent examples include concerns about mammograms and prescreening for prostate cancer. In the political sphere, there has been significant criticism of testing welfare recipients for drug use.

- Provide a brief summary of the article you read. Be sure to point out exactly the nature of the false positives and what cost there is to taking action based on positive results.
- From the article or from other supporting documents, estimate the rate of false positives, the rate of false negatives and the overall rate of prevalence of the condition that is being tested for. Based on your estimates, what is the approximate probability that a positive test is correct?
- Based on what you have read, do you think that current policy should change?

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