Problems in Elementary Probability Theory

Description

This data set contains responses to problems in elementary probability theory observed before and after some instructions (the so-called learning object) were given. Data were collected both in the lab and via an online questionnaire. Of the 1127 participants eligible in the online study, 649 were excluded because they did not complete the first set of problems (p101, …, p112) or they responded too quickly or too slowly. Based on similar criteria, further participants were excluded for the second set of problems, indicated by missing values in the variables b201, …, b212. Problems were presented in random order.

Participants were randomized to two conditions: an enhanced learning object including instructions with examples and a basic learning object without examples. Instructions were given on four concepts: how to calculate the classic probability of an event (pb), the probability of the complement of an event (cp), of the union of two disjoint events (un), and of two independent events (id).

The questionnaire was organized as follows:

Page 1
Welcome page.
Page 2
Demographic data.
Page 3
First set of problems.
Page 4 to 8
Instructions (learning object).
Page 9
Second set of problems.
Page 10
Feedback about number of correctly solved problems.

Usage

data(probability)

Format

A data frame with 504 cases and 68 variables:

  • case a factor giving the case id, a five-digits code the fist digit denoting lab or online branch of the study, the last four digits being the case number.

  • lastpage Which page of the questionnaire was reached before quitting? The questionnaire consisted of ten pages.

  • mode a factor; lab or online branch of study.

  • started a timestamp of class POSIXlt. When did participant start working on the questionnaire?

  • sex a factor coding sex of participant.

  • age age of participant.

  • educat education as a factor with three levels: 1 secondary school or below; 2 higher education entrance qualification; 3 university degree.

  • fos field of study. Factor with eight levels: ecla economics, business, law; else miscellaneous; hipo history, politics; lang languages; mabi mathematics, physics, biology; medi medical science; phth philosophy, theology; psco psychology, computer science, cognitive science.

  • semester ordered factor. What semester are you in?

  • learnobj a factor with two levels: enhan learning object enhanced with examples; basic learning object without examples.

The twelve problems of the first part (before the learning object):

  • p101 A box contains 30 marbles in the following colors: 8 red, 10 black, 12 yellow. What is the probability that a randomly drawn marble is yellow? (Correct: 0.40)

  • p102 A bag contains 5-cent, 10-cent, and 20-cent coins. The probability of drawing a 5-cent coin is 0.35, that of drawing a 10-cent coin is 0.25, and that of drawing a 20-cent coin is 0.40. What is the probability that the coin randomly drawn is not a 5-cent coin? (0.65)

  • p103 A bag contains 5-cent, 10-cent, and 20-cent coins. The probability of drawing a 5-cent coin is 0.20, that of drawing a 10-cent coin is 0.45, and that of drawing a 20-cent coin is 0.35. What is the probability that the coin randomly drawn is a 5-cent coin or a 20-cent coin? (0.55)

  • p104 In a school, 40% of the pupils are boys and 80% of the pupils are right-handed. Suppose that gender and handedness are independent. What is the probability of randomly selecting a right-handed boy? (0.32)

  • p105 Given a standard deck containing 32 different cards, what is the probability of not drawing a heart? (0.75)

  • p106 A box contains 20 marbles in the following colors: 4 white, 14 green, 2 red. What is the probability that a randomly drawn marble is not white? (0.80)

  • p107 A box contains 10 marbles in the following colors: 2 yellow, 5 blue, 3 red. What is the probability that a randomly drawn marble is yellow or blue? (0.70)

  • p108 What is the probability of obtaining an even number by throwing a dice? (0.50)

  • p109 Given a standard deck containing 32 different cards, what is the probability of drawing a 4 in a black suit? (Responses that round to 0.06 were considered correct.)

  • p110 A box contains marbles that are red or yellow, small or large. The probability of drawing a red marble is 0.70 (lab: 0.30), the probability of drawing a small marble is 0.40. Suppose that the color of the marbles is independent of their size. What is the probability of randomly drawing a small marble that is not red? (0.12, lab: 0.28)

  • p111 In a garage there are 50 cars. 20 are black and 10 are diesel powered. Suppose that the color of the cars is independent of the kind of fuel. What is the probability that a randomly selected car is not black and it is diesel powered? (0.12)

  • p112 A box contains 20 marbles. 10 marbles are red, 6 are yellow and 4 are black. 12 marbles are small and 8 are large. Suppose that the color of the marbles is independent of their size. What is the probability of randomly drawing a small marble that is yellow or red? (0.48)

The twelve problems of the second part (after the learning object):

  • p201 A box contains 30 marbles in the following colors: 10 red, 14 yellow, 6 green. What is the probability that a randomly drawn marble is green? (0.20)

  • p202 A bag contains 5-cent, 10-cent, and 20-cent coins. The probability of drawing a 5-cent coin is 0.25, that of drawing a 10-cent coin is 0.60, and that of drawing a 20-cent coin is 0.15. What is the probability that the coin randomly drawn is not a 5-cent coin? (0.75)

  • p203 A bag contains 5-cent, 10-cent, and 20-cent coins. The probability of drawing a 5-cent coin is 0.35, that of drawing a 10-cent coin is 0.20, and that of drawing a 20-cent coin is 0.45. What is the probability that the coin randomly drawn is a 5-cent coin or a 20-cent coin? (0.80)

  • p204 In a school, 70% of the pupils are girls and 10% of the pupils are left-handed. Suppose that gender and handedness are independent. What is the probability of randomly selecting a left-handed girl? (0.07)

  • p205 Given a standard deck containing 32 different cards, what is the probability of not drawing a club? (0.75)

  • p206 A box contains 20 marbles in the following colors: 6 yellow, 10 red, 4 green. What is the probability that a randomly drawn marble is not yellow? (0.70)

  • p207 A box contains 10 marbles in the following colors: 5 blue, 3 red, 2 green. What is the probability that a randomly drawn marble is blue or red? (0.80)

  • p208 What is the probability of obtaining an odd number by throwing a dice? (0.50)

  • p209 Given a standard deck containing 32 different cards, what is the probability of drawing a 10 in a red suit? (Responses that round to 0.06 were considered correct.)

  • p210 A box contains marbles that are green or red, large or small The probability of drawing a green marble is 0.40, the probability of drawing a large marble is 0.20. Suppose that the color of the marbles is independent of their size. What is the probability of randomly drawing a large marble that is not green? (0.12)

  • p211 In a garage there are 50 cars. 15 are white and 20 are diesel powered. Suppose that the color of the cars is independent of the kind of fuel. What is the probability that a randomly selected car is not white and it is diesel powered? (0.28)

  • p212 A box contains 20 marbles. 8 marbles are white, 4 are green and 8 are red. 15 marbles are small and 5 are large. Suppose that the color of the marbles is independent of their size. What is the probability of randomly drawing a large marble that is white or green? (0.15)

Further variables:

  • time01, …, time10 the time (in s) spent on each page of the questionnaire. In the lab branch of the study, participants started directly on Page 2.

  • b101, …, b112 the twelve problems of the first part coded as correct (1) or error (0).

  • b201, …, b212 the twelve problems of the second part coded as correct (1) or error (0).

Source

Data were collected by Pasquale Anselmi and Florian Wickelmaier at the Department of Psychology, University of Tuebingen, in February and March 2010.

Examples

library("pks")

data(probability)

## "Completer" sample
pb <- probability[!is.na(probability$b201), ]

## Response frequencies for first and second part
N.R1 <- as.pattern(pb[, sprintf("b1%.2i", 1:12)], freq = TRUE)
N.R2 <- as.pattern(pb[, sprintf("b2%.2i", 1:12)], freq = TRUE)

## Conjunctive skill function, one-to-one problem function
sf1 <- read.table(header = TRUE, text = "
  item cp id pb un
     1  0  0  1  0
     2  1  0  0  0
     3  0  0  0  1
     4  0  1  0  0
     5  1  0  1  0
     6  1  0  1  0
     7  0  0  1  1
     8  0  0  1  1
     9  0  1  1  0
    10  1  1  0  0
    11  1  1  1  0
    12  0  1  1  1
")

## Extended skill function
sf2 <- rbind(sf1, read.table(header = TRUE, text = "
  item cp id pb un
     2  0  0  0  1
     3  1  0  0  0
     6  0  0  1  1
     7  1  0  1  0
    12  1  1  1  0
"))

## Delineated knowledge structures
K1 <- delineate(sf1)$K
K2 <- delineate(sf2)$K

## After instructions, fit of knowledge structures improves
sapply(list(N.R1, N.R2), function(n) blim(K1, n)$discrepancy)
[1] 1.2985507 0.8956522
sapply(list(N.R1, N.R2), function(n) blim(K2, n)$discrepancy)
[1] 1.1623188 0.8231884