Question
As a broad generalization (which you can verify empirically), statisticians tend to have shy personalities more often than economists do let's quantify this observation by assuming that 80% of statisticians are shy but the corresponding e among economists is only 15%. (Fact 2) Conferences on the topic of econometrics are almost exclusively attended by economists and statisticians, with the majority of participants being economists let's quantify this fact by assuming that 90% of the attendees are economists (and the rest statisticians). Suppose that you (a physicist, say) go to an econometrics conference. you strike up a conversation with the first person you (haphazardly) meet, and find that this person is shy. The point of this problem is to show that the (conditional) probability p that you're talking to a statistician is only about 37%, which most people find surprisingly low, and to understand why this is the right answer. Let St = (person is statistician), E = (person is economist), and Sh = (person is shy). (a) Using the St, E and Sh notation, express the three numbers (80%, 15%, 90%) above, and the probability we're solving for, in unconditional and conditional probability terms. (b) Briefly explain why calculating the desired probability is a good job for Bayes's Theorem. (c) Briefly explain why the following expression is a correct use of Bayes's Theorem in odds form in this problem.[P(St|Sh)/P(E|Sh)] = [P(St)/P(E)] middot [P(Sh|St)/P(Sh|E)] (1) = (2) middot (3)
Explanation / Answer
a) P(St) = 10%
P(E) = 90%
P(Sh | St) = 80%
P(Sh | E) = 15%
P(Sh) = P(St and Sh) + P(E and Sh)
= 10x0.8 + 90x0.15
= 21.5%
b) Using Baye's thorem, we can estimate the probability under a particulat circumstance, that is under a given condition. So, our desirable probabilities are usually having a condition attached to it. We often need to find the probability of something happening, when something have already happened. In such circumstances, we use Baye's theorem and we can calculate the conditional probability.
If you meet a person who is shy, the probability of him being a statistician is given by P(St | Sh)
= P(St and Sh) / P(sh)
= 0.1x0.8/0.215
= 0.3721 = 37%
c) (1) = [P(St | Sh) / P(E | Sh)] = [0.37/(0.9x0.15/0.215)] = 0.59
(2) = P(St) / P(E) = 0.1/0.9 = 0.11
(3) = [P(Sh | St)] / [P(Sh | E)] = [0.8 / 0.15] = 5.33
(2).(3) = 5.33x0.11 = 0.59
(1) = (2).(3)
