Question

Suppose that 80 percent of all statisticians are shy, whereas only 15 percent of all economists are shy. Suppose also that 90 percent of the people at a large gathering are economists and the other 10 percent are statisticians. If you meet a shy person at random at the gathering, what is the probability that the person is a statistician?

Short answer

The probability that the person is a statistician is 0.372093

Step-by-step solution

Given information

Suppose that 80 percent of all statisticians are shy, whereas only 15 percent of all economists are shy. Suppose also that 90 percent of the people at a large gathering are economists and the other 10 percent are statisticians.

Calculating the probability

Let define events,

A: Gathering are Statistician

B: Gathering are Economics are shy

S: The person is shy

\(\begin{aligned}{}\Pr \left( A \right) &= 0.10\\\Pr \left( B \right) &= 0.90\\\Pr \left( {S|A} \right) &= 0.80\\\Pr \left( {S|B} \right) &= 0.15\end{aligned}\)

If one meet a shy person at random at the gathering, what is the probability that the person is a statistician obtained as:

\(\begin{aligned}{}i.e,\Pr \left( {A|S} \right) &= \frac{{\Pr \left( {A \cap S} \right)}}{{\Pr \left( S \right)}}\\ &= \frac{{\Pr \left( A \right)\Pr \left( {S|A} \right)}}{{\Pr \left( A \right)\Pr \left( {S|A} \right) + \Pr \left( B \right)\Pr \left( {S|B} \right)}}\\ &= \frac{{0.10 \times 0.80}}{{0.10 \times 0.80 + 0.90 \times 0.15}}\\ &= 0.372093\end{aligned}\)

The probability that the person is a statistician is 0.372093.