Source: exercise in Maddala's undergraduate text

Concepts Introduced and Defined:

[sample space] [event] [probability] [independent events].

Setup:

You are interested in surveying students about drug use. You fear people will not tell the truth. You cleverly design the following method to ask questions but protect privacy.

You place in a box:

4 White balls

3 Red balls

4 Blue balls

Students pick a ball that you cannot see. You give them the following instructions:

If Blue, answer trufthfully, "Have you ever used cocaine?"

If Red, answer "No."

If White, answer "Yes."

Question: 40 Percent of the students say "Yes." What proportion of your sample has tried cocaine?

Answer:

Define x

proportion of class that has tried drugs

Sample Space

S = {(R,T), (B,T), (W,T), (R,NT), (B,NT), (W,NT)}

Events that can be observed by the surveyor:

Y = {(B,T),(W,T),(W,NT)}

N = {(B,NT),(R,T),(R,NT)}

Y and N are mutually exclusive and exhaustive events:

Y U N = S

Y AND N = 0 (empty set)

P(Y) + P(N) = 1

Make the following Assumptions

Assumption 1:

Ball choice and druge use are independent events:

P(ball,drug) = P(ball)*P(drug)

Assumption 2

Each ball in the box is equally likely

P(B) = 4/11

P(R) = 3/11

P(W) = 1-P(R)-P(B) = 4/11

Assumption 3:

Students follow the instructions:

P(Yes) = P(B,T) + P(W,T) + P(W,NT)
=(4/11)*x + (4/11)*x + (4/11)*(1-x)
= (4/11) + (4/11)*x

so x = P(Yes)(11/4)-1

The question tells us

P(Yes) = .40 = (4/11) + (4/11)*x

x = (11/4)*(4/10)-1 = 11/10 - 1 = 1/10

So 1/10 of the students have tried cocaine

Other issues:

Small sample variation, estimation, inference