## A. How many of your classmates do drugs?

### 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?"

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

#### 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

#### 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:

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

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