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- 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

Document Last revised: 1997/1/5