2. Applications for funding from the Centralian Research Council can be either p
ID: 3375433 • Letter: 2
Question
2. Applications for funding from the Centralian Research Council can be either poor, fair or good. The assessment process is not perfect. Specifically the probabilities of grants being rated poor, fair or good given that they are poor, fair or good are given in the table below. rating p 0.5 0.3 0.2 qualityf 0.3 0.4 0.3 g 0.2 0.2 0.6 Applications are funded if and only if they are rated as good (a) Assume that an arbitrary application is poor, fair or good with probability 0.2, 0.4 and 0.4 respectively. What is the probability that it will be funded? (b) What is the probability that a funded application was actually poor? (c) Carefully stating your reasons, explain whether the events (p), that an arbitrary application is poor, and (rf), that it is rated fair, are independent. 7 marks]Explanation / Answer
(a) Here,
Pr(Poor) = 0.2
Pr(Fair) = 0.4
Pr(Good) = 0.4
Pr(It will be funded) = Pr(Poor) * Pr(Good when really poor) + Pr(Fair) * Pr(Good when really Fair) + Pr(Good) * Pr(Good when really fair)
= 0.2 * 0.2 + 0.4 * 0.3 + 0.4 * 0.6 = 0.4
(b) A funded application is poor
Pr(Poor l Funded application is really poor) = (0.2 * 0.2)/0.4 = 0.10
(c) If two events are independent then
Pr(Poor application) * Pr(rated fair) = Pr(Poor appication rate fair)
Pr(Poor application) = 0.2
Pr(rated fair) = (0.2 * 0.3 + 0.4 * 0.4 + 0.4 * 0.2) = 0.3
Pr(Poor application) * Pr(Rated fair) = 0.2 * 0.3 = 0.06
Pr(Poor application rated fair) = 0.2 * 0.3 = 0.06
so yes these events are independent in nature.
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