Question

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. Briefly explain what key assumption is necessary to validly bring probability into the solution of this problem?
b. 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.
c. Briefly explain why calculating the desired probability is a good job for Bayes's The- orem

Answer 1

Explanation:

Given that :

St = the event that person is statistician

E = the event that person is Economist

Sh = the event that person is Shy

a. Briefly explain what key assumption is necessary to validly bring probability into the solution of this problem?

St and E are exclusive events since a person cannot be both statistician and economist.

Key Assumptions:

P(St) + P(E) = 1

Also;

P (St ∩ E) = ∅

b. 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.

Given that :

80 % (0.8) of the statisticians are shy and also 15% (0.15) of the economist too are shy; Then :

`P(Sh|st) = 0.8`

`P(Sh|E) = 0.15`

In the conference; it is stated that there are 90% economist ; Therefore:

P(E) = 0.9

P(St) = 0.1

c) Briefly explain why calculating the desired probability is a good job for Bayes's The- orem

From the foregoing; we knew the probability of
`P(E), P(st) , P(Sh|St) , P(Sh|E)` and asked to show that P(st|sh) = 0.37 ; Then using Bayes Theorem; we have:

`P(St|Sh) = (P(St)*P(Sh|St))/(P(E)*P(Sh|E)*P(St)*P(Sh|St)) = 0.37`

`P(St|Sh) = (0.1*0.8)/(0.1*0.8+0.9*0.15) = 0.37`

`P(St|Sh) = (0.08)/(0.215) = 0.37`

As illustrated above; the required probability was determined using Bayes Theorem; Thus, calculating the desires probability is a good job for Bayes's The- orem.