This document contains instructions, some answers, comments, tips and tricks, and further explanations for the exercises.

Exercise 5.1; explanation

Define: POS = “LIAR”; this is a choice, it’s also possible to define “LIAR” as a negative.

Based on performing the test a couple of times, a so called Confusion Matrix can be created.

ACTUAL
POSITIVE NEGATIVE TOTAL
PREDICTED POSITIVE TP FP PRED.POS
NEGATIVE FN TN PRED.NeG
TOTAL POS NEG N (sample size)

In the exercise it is not clear what is meant by ‘the probability of a false positive’; it could at least mention three probabilites (which three?).
However there is a definition for the False Positive Rate FPR, this is the rate of the Actual Negatives predicted (or classified) as Positives. Or as a formula: \[ FPR = \frac{FP}{NEG}\]
So if FPR = .08 it means that using the lie detector many many times, on average the lie detector classifies 8% of non-liars as liars.

Remark
In the literature and on the internet lots of examples of Confusion Matrices (CF’s) can be found. Be aware that there is no uniform structure for a CF. Sometimes the predicted values are in the rows and the actual values in the columns, as in the CF above. Sometimes it is the other way around. Sometimes the first mentioned categories are the Positives, and sometimes the Negatives are mentioned first.
For instance, a Confusion Matrix may look like this:

PREDICTED
POSITIVE NEGATIVE TOTAL
ACTUAL POSITIVE TP FN PRED.POS
NEGATIVE FP TN PRED.NeG
TOTAL POS NEG N (sample size)


EXERCISE
Fill in TP, TN, FP, FN, POS, NEG, PRED.POS en PRED.NEG in the correct cell in the CF below.

PREDICTED TOTAL
NEGATIVE POSITIVE TOTAL
ACTUAL NEGATIVE
POSITIVE
TOTAL N (sample size)


Exercise 5.2; remarks

Read the text carefully and you see that a Positive is defined as a woman who has breast cancer.

Definition False Negative Rate, the rate of Actual Positives classified as Negatives.
\[ FNR = \frac{FN}{POS}\] FNR = 0.10 means that on the long run of all the women who have breast cancer, 10% is classified as not having breast cancer.


Exercise 5.12; short answers

  1. A roulette wheel doesn’t have a memory; the chances do not change after five consecutive reds. In other words, the outcomes of different spins of the wheel are independent.
  2. Because the cards are dealt without replacement, the probability of receiving a red card after receiving one or more red cards straight, decreased.