Abstract
The compulsory university entrance tests in Andalucía (Spain), which traditionally include a probability problem, often determine whether the student is admitted to study in his or her desired university program. Consequently, it is important to ensure that the content of the test is directly related to the content included in the high school curriculum. The aim of this research was to investigate the distribution of the main variables characterizing the probability problems posed in these tests in Andalucía. Specifically, we examined all the problems posed to students in the period 2003–2014 (n = 144 problems). We considered the following variables: type of experiment and sample space considered, type of probability that should be computed, theorems or properties needed to find the solution, format of the data, and context. The results of the analysis reveal the potential difficulty of the problems, as well as the relevance given to conditional probability in the tests, in comparison to other curricular contents.
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Acknowledgements
Research Projects EDU2016-74848-P (AEI, FEDER), and EDU2013-41141-P (MINECO), and Research Group FQM126 (Junta de Andalucía).
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Appendix: Examples of Problems
Appendix: Examples of Problems
Example 1
Consider two events A and B:
-
(a)
Describe, using set operations, the following events: (1) None of the two events happen; (2) At least one of the two events happen; (3) B happens and A does not happen.
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(b)
Suppose it is known that P(A ) = 0.5, P(B) = 0.5, and P(A |B) = 0.3. Find P(A \( \cup \)B).
Example 2
55% of the Spanish population are women, 23% of which use travel to work by car. The probability that a person (either man or woman) travels by car to work is 0.52.
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(a)
We select a man at random, what is the probability that this man travels to work by car?
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(b)
We select a person at random and this person travels to work by car. Compute the probability that this person is a woman.
Example 3
Blanca and Alfredo write a vowel at random on different pieces of paper.
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(a)
Determine the sample space in this experiment.
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(b)
Find the probability that they write different vowels.
Example 4
There are only physics and mathematics books in a library and the books are written either in English or in Spanish. 70% of the books are physics books, 80% of the books are written in Spanish, and 10% of the books are mathematics books written in English.
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(a)
What percentage of books are physics books written in Spanish?
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(b)
We select a physics book at random, what is the probability that the book is written in Spanish?
Example 5
Consider the random experiment consisting of flipping a fair coin three times.
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(a)
List the associated sample space and compute the probabilities of the elementary events.
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(b)
Let A be the event : “obtain at least one head,” and let B be the event : “obtain a head in only one of three throws.” Calculate P(A ) and P(B). Are A and B independent?
Example 6
Two urns A and B contain coloured balls. Urn A contains 4 white balls and 3 red balls. Urn B contains 5 white balls, 2 red balls, and 1 black ball. A die is rolled. If the result is 1, 2, 3 or 4, a ball is randomly selected from urn A ; if the result is 5 or 6, a ball is randomly selected from urn B.
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(a)
Find the probability of getting a red ball.
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(b)
Find the probability of getting a black ball.
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(c)
If a white ball was obtained, find the probability that a 5 or 6 was rolled on the die.
Example 7
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(a)
Let A and B be two events from the same sample space . Suppose it is known that P(A ) = 0.5, P(B) = 0.4, and P(A \( \cup \) B) = 0.8. Determine P(A |B).
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(b)
C and D are two independent events from the same sample space . P(C) = 0.3 and P(D) = 0.8. Determine P(C \( \cup \) D).
Example 8
A student travels to school by bus on 80% of school days and travels by car the remaining days. When he travels by bus, he is late 20% of the time, but when he travels by car he is on time only 10% of the time. A day is selected at random.
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(a)
Find the probability of arriving on time and travelling by bus.
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(b)
Compute the probability of arriving late.
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(c)
If the student arrived on time, compute the probability that he had travelled by bus.
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Batanero, C., López-Martín, M., Arteaga, P., Gea, M.M. (2018). Characterizing Probability Problems Posed in University Entrance Tests in Andalucia. In: Batanero, C., Chernoff, E. (eds) Teaching and Learning Stochastics. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-72871-1_7
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