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

Answer:

Step-by-step 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:

$P(St|Sh) = \frac{P(St)*P(Sh|St)}{P(E)*P(Sh|E)*P(St)*P(Sh|St)} = 0.37$

$P(St|Sh) = \frac{0.1*0.8}{0.1*0.8+0.9*0.15} = 0.37$

$P(St|Sh) = \frac{0.08}{0.215} = 0.37$

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.