1 THE NATURE OF THE QUANTITIES IN A CONDITIONAL PROBABILITY PROBLEM. ITS INFLUENCE IN THE PROBLEM RESOLUTION. M. Pedro Huerta (Universitat de València), Mª Ángeles Lonjedo (IES Montserrat) ABSTRACT In order to solve verbal conditional probability problems, students are involved in a process in which we can identify several steps or phases. One of them is that of the translation from the text of the problem, generally written in a vernacular language, to that of mathematics. In translating sentences, students should recognize events and probabilities. But, in much of those problems data are not explicitly mentioned in terms of probability. In this case, students can solve these problems with the help of arithmetical thinking and not necessarily with the help of probabilistic reasoning. Other works (Ojeda 996, Huerta-Lonjedo 3) already talked about this, but not adequately. In this piece of work we investigate, through an exploratory study of 66 students from different school levels, the extent to which the nature of quantities in conditional probability influences the way in which students solve these problems. INTRODUCTION We use the term problem in a Puig (996) sense, that is, any problematic situation in a scholar context. Probability problem is a problem in which the question is about the probability of an event and conditional probability problem is a probability problem in which at least a conditional probability is involved, either as a data and question or both. In this report we are considering verbal conditional probability problems, that is conditional probability problems written in a vernacular language. In addition to the nature of data, there are, as we know, some others factors that also have an influence on problem solving of conditional probability tasks. One of these factors is not necessarily the previous knowledge about the relationships between probabilities but, for example, the identification of the events and their probabilities. The prior identification of events and the corresponding assignment of their probabilities have to do with semiotic and semantic aspects as well as the right correlation between data and events. In this piece of work we are not going to deal with this issue but with the nature of quantities in problems, and its influence on the problem solving process. When data in a conditional probability problem are expressed in terms of frequencies, percentages or rates, students do not necessarily interpret them as probabilities. Consequently, the relationships between probabilities are not used when students are solving problems, at least in an explicit way. However, this does not mean that no student can solve these problems. Of course, there are students that succeed in solving these problems, but they mainly use arithmetic thinking and not probabilistic thinking. It is only at the end of the problem solving process that students answer the question in terms of a required probability, usually by assignment methods. In this
2 paper, we will show the results of an exploratory study with 66 students from different school levels solving conditional probability problems in which the data are not always explicitly mentioned as probabilities in the text of problems. THE NATURE OF THE QUANTITIES IN A CONDITIONAL PROBABILITY PROBLEM. It has been through the study of conditional probability problems in textbooks, that we have noticed that data in most of them are not always expressed in terms of probabilities. Previous studies (Ojeda 996; Huerta, Lonjedo 3; Lonjedo, 3) show that, because of the way data are often mentioned and expressed in the text of problems, the problems could be solved just by using numeric thinking. Of course, we have also very often observed students using arithmetic thinking when solving some conditional probability problem and not necessarily probabilistic thinking. This is because data are not being interpreted consciously as probabilities and consequently, students do not need to use the relationships between probabilities to solve the problem. It is only at the end of the problem solving process that students try to express their answer in terms of the required probability. This difference in students approach when solving probability problems, using arithmetical thinking or probabilistic thinking, let us classify probability problems in probability problems by assignment and probability problems by calculation, and also conditional probability problems. From this point of view, school conditional probability problems found in textbooks could be classify as assignment problems or as calculation problems. So, a conditional probability problem will be classified as calculation problem if data in the problem are interpreted as probabilities and, consequently the relationships between probabilities are needed in order to answer the question in problem. Nevertheless, past teaching experience in probability demonstrates the opposite. Teachers and textbooks do not often start from this empirical fact to reach probabilistic thinking. They usually present problematic situations that demand this type of thinking, not being necessarily requested by the solution. Here we have an example. (Cuadras C.M, (983) Problemas de Probabilidades y Estadística, Vol : Probabilidades p.55) Dos máquinas A y B han producido respectivamente, y piezas. Se sabe que A produce un 5% de piezas defectuosas y B un 6%. Se toma una pieza y se pide: a) Probabilidad de que sea defectuosa. b) Sabiendo que es defectuosa, probabilidad de que proceda de la primera máquina. (Two machines A and B produce respectively and pieces. It is known that machine A produces 5% of faulty pieces and machine B produces 6% of faulty. Resolution of the schoolbook
3 pieces. If you took a piece at random calculate: a) The probability that this piece would be faulty. b) Knowing that the piece is faulty, the probability that it is made by machine A ) The solution of the problem in textbook can be seen in Figure. The book considers this problem as a problem of calculation, in concordance with the lesson the problem is, in the unit Total Probability Theorem and Bayes Theorem. However, some students (Lonjedo, 3) solved problems similar to that in both data nature and in data structure, as we show below: Si tenemos piezas de la máquina A y el 5% son defectuosas: tenemos 5 piezas defectuosas de las de A. Si tenemos piezas de la máquina B y el 6% son defectuosas: tenemos piezas defectuosas de las de B. En total, de 3 piezas de las dos ) máquinas, 5+=7 son defectuosas. Luego la probabilidad de ser defectuosa es: 7 de 3 o.56 Para la segunda cuestión, tenemos 7 piezas defectuosas, de donde 5 vienen de la máquina primera, luego la probabilidad pedida es de: 5 de 7 o.94 (If we have pieces from the machine A and 5% are faulty, in pieces we have 5 faulty pieces. If we have pieces from machine B and 6% are faulty, in pieces we have faulty pieces. In total, among 3 pieces we have 7 (5+) faulty pieces. So, if we take a piece at random, the probability that it will be faulty is 7 de 3 o 7/3 o We have 7 faulty pieces, 5 of them are from the machine A and from de the machine B. So, if we know that the piece is faulty the probability it is made by machine A is 5 out of 7 or 5/7 or.94). DIFFERENT NATURE OF DATA The analysis of some textbooks allows us to classify conditional probability problems according to the nature of the data in the text of the problem. So, we can distinguish different types of them: Data expressed in probability terms. If quantities are expressed in terms of probability, they quantify the probability of a certain event A by a number p (A) [,], as in the following example: A B.4. nob.5 noa Total (Matemáticas 4t ESO, Opción B, Editorial Ecir, página 4, problema 43). Completa la següent taula de contingència. A partir de la taula, confecciona un diagrama d arbre i determina P(B/A), P(noB/A), P(B/noA) i P(noB/noA). (Complete the next contingency table. From this table build a tree diagram and calculate p (B A), p (nob A), p (B noa) and p (nob noa) ) Total Here, in this problem, p(a B) must be calculated by: p(a B)=p(A B)/p(B), that is, only by using probabilistic thinking the relationship between probabilities. Data expressed in absolute frequencies. When in a conditional probability problem data are expressed in terms of absolute frequencies, they express the frequency of the objects that satisfy certain characteristics. From a mathematical point of view, frequency can be seen as a cardinal number associated to the set that represents those objects. So, the quantity in a problem given as a frequency must be used with that P(noB A) means P( B A)
4 meaning. Consequently, p (A B) must be obtained by comparing two numbers and expressing it, for example, by this:, p (A B) = n(a B)/n(B), that is, by using arithmetical thinking and not relationships between probabilities. On the other hand, because A B is not an event, we cannot consider a set that represents it. So, data referring to a conditional probability could never be expressed in terms of absolute frequencies. If we do so, the only meaning that one can associate to that data will be that of a cardinal number associated to an intersection event. Here we have an example: (Editorial Santillana, Matemátias Cou opciones C y D, Daniel Santos Serrano, (p.48, problema 9) En un grupo de 5 individuos se pasó un test de inteligencia y se midió su rendimiento académico. Los resultados fueron como sigue (tabla de contingencia). Considerando que A es ser superior en inteligencia y B es tener rendimiento alto, averiguar :a)si A y B son independientes. b) Si se selecciona al azar un alumno con rendimiento alto, cuál es la probabilidad de que sea superior en inteligencia? (An intelligence test was administrated to a group of 5 students to assess their academic performance. The results of that test were as follows (contingency table). Let A be having higher intelligence test and B having a higher academic performance, Answer the questions: a)are A and B independent events? b) If we randomly choose a student with higher performance at school, what is the probability that he/she was higher intelligence?) Data expressed in terms of rate. When quantities are shown in terms of rates, data are implicitly expressed in terms of probability and it is up to the solver to decide whether to translates the rates to probabilities or not. Here, we have two examples. The first one shows two rates as data and the second one shows data in percentages: Engel, L enseignement des probabilités et de la statistique, volumen (975) (problema 9, p.7) (France; CEDIC) En Sikinie, un homme sur et une femme sur 88 sont daltoniens. Les fréquences des deux sexes sont égales. On choisit une personne au hasard et on découvre qu elle est daltonienne. Quelle est la probabilité pour que ce sois un homme? (In Sikinie one man out of and one woman out of 88 are affected with Daltonism. The frequencies of men and women are the same. A person is chosen at random and it is known that he/she is affected with Daltonism. What is the probability that this person is a man?) Grupo Cero, Borrás, Carrillo, D Opazo, Morata, Puig,, Matemáticas de Bachillerato curso, (98), (problema 6, p. 7.)(Barcelona: Teide). En el proceso de fabricación de circuitos impresos para radio transistores se obtiene, según demuestra la experiencia de cierto fabricante, un 5% de circuitos defectuosos. Un dispositivo para comprobar los defectuosos detecta el 9% de ellos, pero también califica como defectuosos al % de los correctos. Cuál es la probabilidad de que sea correcto un circuito al que el dispositivo califica como defectuoso? Cuál es la probabilidad de que sea defectuoso un circuito calificado de correcto? (Experience shows that during the process of making circuits for radio transistors 5% of them faulty. A device that is used to find out which are faulty, detects 9% of the faulty ones, but also qualifies % of faulty circuits as correct. What is the probability that a circuit is correct if the device says that it is faulty? What is the probability that a faulty circuit is qualified as correct?) Rendimiento académico Alto Bajo Superior 8 Inferior Data expressed in combined terms. There are certain conditional probability problems in textbooks where data are not all expressed in only one sense, like in the examples we show above, but by combining more than one sense. Thus, in these problems we can find data explicitly expressed both in terms of probability and percentage, probability and rates, or rates and frequency. In the following problem, for example, data as percentages combines more than one sense.
5 Santos, D., (988), Matemáticas COU, Opciones C y D, Madrid: Santillana. p.48, problema, En un curso el porcentaje de aprobados en Historia (A) es 6 %. Para Matemáticas (B) es del 55 %. Sabiendo que p(b/a) = 7 %, cuál es la probabilidad de que, escogido al azar un alumno, resulte no haber aprobado ninguna de las dos asignaturas? (In a high school class, the percentage of students that succeeded in History (A) was 6%. In Mathematics (B) it was 55%. Knowing that p(a B)=7%, what is the probability that a student chosen at random did not succeed in either topic?) So, we have observed through the examples we have shown how data in conditional probability problems are not always expressed in probability terms or with the same sense. So, when this happens, the solver should have the ability to interpret them or not according to the desired meaning in the problem. Based on that, the problem solving process can imply either numerical or probabilistic thinking, depending on how the solver interprets the data. THE EMPIRICAL STUDY One of the objectives of our study was to explore how students solved conditional probability problems when data in problems satisfied certain conditions in their structure and nature. Mainly, we were interested in exploring what kind of thinking arithmetical or probabilistic students used in solving the problems, in relation to the data structure and nature. THE TEST To complete this study, we prepared a collection of sixteen conditional probability problems with a similar data structure, varying the nature of the data and the context. So, all problems had three pieces of data explicitly mentioned in the problem text, but they were expressed either in terms of percentages or in terms of probability. On the other hand, all sixteen problems have an arithmetic solution and are by isomorphisms. The isomorphism of the problems corresponds to the structure of data. For example, in problem, one can interprete data as follows p(a B)=3%, p(a ~B)=3% and p(b ~A)=4%, but in its isomorphic problem, problem n.9, data are explcitly mentioned as probabilities p(a B)=.3, p(a ~B)=.3, p(b ~A)=.4. Both problems, however, share the same question calculate p(b) and the same context. So, that collection of problems contained problems from to 8 with data in percentages and problems 9 to 6, with data explicitly expressed in terms of probability. That is, we have eight pairs of problems that are by isomorphisms: - 9, -, 3-, 4-, 5-3, 6-4, 7-5, and 8-6. These problems can be found in the appendix at the annex and only in a Spanish language version 3. Considering the time limitations students had, we asked each student to solve a total of four problems (two from the -8 list and two from the 9-6 list), in order to have as many solved problems as possible and from all school levels. THE STUDENTS That is, an extra data as an unknown quantity is not required in order to solve problem. 3 Because the semantic and semiotic aspects are influential factors, we are not sure that in translating the text of problems from the Sapnish language to the English language we can retain the same meanings. So, because we are not competent in English language we prefer not to tanslate the problems.
6 The sample of students that took part in the study was a total of 66 students distributed as follows: EFM: students of the Didactics of Mathematics subject at the Math s College. 38 BT-C: students of nd year of high school, studying the option of technology and science (7-8 years old). 6 BCS-H: students of nd year of high school, studying the option of social science and humanities (7-8 years old). 38 BT-C: students of st year of high school, studying the option of technology and science (6-7 years old). 37 BCS-H: students of st year of high school, studying the option of social science and humanities (6-7 years old). 7 4ESO: students of 4th year of Compulsory Secondary School (5-6 years old). That is, we had students from the Faculty of Mathematics, 54 students at their last year before entering the University and with different competence in mathematics, 75 students at two years before entering the University and also with different competence in mathematics and finally 7 students at the end of the compulsory secondary school. The test was administered during student s regular class time at their high school or college. ANALYSIS OF RESULTS The results shown in the tables below have been organized according to the following variables: the nature of the data, the number of students that have attempted to solve each problem; the number of students that succeeded in solving each problem; the distribution of the number of students that succeeded depending on their school level; the number of students that did not answer the specific problem or similar ones; and depending on the reasoning used in problem solving, the distribution of the number of students that succeeded in probability assignment or probability calculation. Tables and organize information about those students that succeeded in solving the problems and also about the number of students that did not attempt to solve the problems or similar blank. So, information about those students that did not complete successfully the problems is not reported here. In the process of solving the problems we could observe mistakes and misunderstandings of a different nature. Problem P P P3 P4 P5 P6 P7 P8 P9 P P P P3 P4 P5 P Data nature % % % % % % % % Prob. Prob. Prob. Prob. Prob. Prob. Prob. Prob. Sample
7 Succeeded EFM 4 BT-C 3 6 BCS-H 3 BT-C 8 BCS-H 3 4ESO Blank Probability Assignment Probability Calculation. (EFM) (EFM) Table : Number of students succeeded in solving the problems and their distribution depending on school level and type of reasoning used. Problem. P P P3 P4 P5 P6 P7 P8 P9 P P P P3 P4 P5 P6 Nature of Data % % % % % % % % Prob. Prob. Prob. Prob. Prob. Prob. Prob. Prob. Succeeded Blank Probability Assignment Probability Calculation (EFM) (EFM) Table Percentages of students succeeded and blank and, about those that succeeded, percentages of students that used a type of thinking. Tables & show similar results, with table results specified in absolute frequencies while table results in percentage. It should be noted that for problems 9 to 6 in both tables, where data are expressed in terms of probability, the percentage level of correct solutions is lower than for the first 8 ones. On the other hand, we would like to point out that when data are shown in terms of probability, the number of students
8 trying to solve the problem decreases 4 ; we can see that the percentage of problems left blank or similar is higher in the last 8 problems, except for the problem P where the high percentage of students that left the problem blank stands out. So, we can observe how the nature of the data is an influential factor in the problem s solution process. That is, students that succeeded solved conditional probability problems as a problem of probability assignment when data were expressed in percentages or as a probability calculation when data were expressed in terms of probability. The summary table below gathers those columns that give evidence of the solution process of the problems, the nature of their data and those that have been successfully completed: Problem P P P3 P4 P5 P6 P7 P P P3 P4 P5 Nature of Data % % % % % % % Prob. Prob. Prob. Prob. Prob. Assignment 87 5(EFM) Probability Calculation (EFM) (EFM) 75(EFM) (EFM) Table 3: Percentages of the types of resolutions for those problems successfully completed. One of the features from the table 3 is that in some cases it contradicts the assertion that data expressed in percentage favors the solution of the problem by assignment. We can see, for example, that in problems 6 and 7; the majority of those students that successful solved the problems used the probability calculation. However, all the students in this sample belong to the Math s College, and are knowledgeable of the theory of probability. We can also notice that in some cases, when data are expressed in terms of probability, some students translate those terms into percentages and solve the problem by using arithmetic thinking and assigning probability at the end of the solution process. This is the case in problem. One student from thebcs-h group who translated data expressed in terms of probability into percentages. When solving problem 4, another student, from the BT-C group, used the same process of translation. The eight problems with the data shown in percentages and the number of students that succeeded in solving them can be seen at table 4: Problem P P P3 P4 P5 P6 P7 P8 Succeeded According to the official curriculum of students for st and nd year at high school studying social science-humanities option, some conditional probability knowledge has been provided. However, this does not always happen. We cannot assure that these students belonging to these courses were knowledgeable in the subject. Consequently, problems which data were expressed in terms of probability are not attempt to be solved very frequently.
9 Blank Table 4: Percentages of those variables that give evidence of the successfulness or failure on the resolution of a problem The results of problem P8 are coherent with the enunciation of the problem and the competence level of the students. The two students from the Math s College, theoretically provided with a good knowledge of the theory of probability, tried to solve it by using wrong formulas. The high percentage of answers left blank or similar is basically due to the way these data are presented: p (B A)=7%. With refer to problem, also with a very high percentage left blank or similar, % correspond to 8 blank answers and 3 with unfinished solution. We do not understand the reason why this happens because the enunciation is similar to problems & 3, where the percentages with a blank response are not so relevant. Moreover, if we observe the results of its isomorphic problem, problem 9, there are a % of left in blank. CONCLUSIONS We suspect that apart from the nature of the data, there are some other factors that have a direct effect on the way students approach a conditional probability problem. These factors are not necessarily related to the knowledge of the relationships between probabilities. The nature of the school s probability data problems is influential in the classification of the problems: problems of probability assignment or problems of probability calculation. The problems solved by the students who took part in this research could be classified as problems of probability assignment. Most of these students did not understand data as probabilities and consequently would never use the relations between probabilities to calculate the probabilities requested in the problems. The data in these problems are presented in percentages and students solve them by using numerical thinking. However, those problems where data are presented in terms of probability are not just less elaborated by the sample students but also have been mostly solved by using the probability calculation rules. The numerical data shown in the selected probability problems, acquire some meaning to the students when they are expressed in terms of percentages. So, in this case, when quantities have certain meaning for students, they can produce new quantities that can also be relevant for the solution of the problem, thus facilitating the solving problem process. However, when quantities are expressed in terms of probabilities they do not make feasible that production of new quantities, mainly if one is not competent with the relationships between probabilities or with formulas. Consequently, we will not be saying anything new (Ojeda, 996) if we continue believing in the solution of probability problems where data suggest a focus on probability as a frequency before this is shown in a formal way. This applies not only to solving probability problems, but also conditional probability problems. If those conditional probability problems were focused in the way we suggested, it would
10 allow us their inclusion in arithmetic lessons or the use of rates and proportions for their solution, as prior step to teaching rules or formulas. REFERENCES Huerta, M. Pedro (3) Curs de doctorat en Didáctica de la probabilitat. Departament de Didàctica de la Matemàtica. Universitat de València (documento interno). Huerta, M. Pedro, Lonjedo, Mª Ángeles (3) La resolución de problemas de probabilidad condicional. In Castro, Flores at alli (eds), 3, Investigación en Educación Matemática. Séptimo Simposio de la Sociedad Española de Investigación en Educación Matemática. Granada. Lonjedo, Mª Ángeles, (3) La resolución de problemas de probabilidad condicional. Un estudio exploratorio con estudiantes de bachiller. Departament de Didàctica de la Matemàtica. Universitat de València (Memoria de tercer ciclo no publicada) Lonjedo, Mª Ángeles, Huerta, M. Pedro, (4) Una clasificación de los problemas escolares de probabilidad condicional. Su uso para la investigación y el análisis de textos. In Castro, E., & De la Torre, E. (eds.), 4, Investigación en Educación Matemática. Octavo Simposio de la Sociedad Española de Investigación en Educación Matemática, pp 9-38, Universidade da Coruña. A Coruña. Ojeda, A.M. (996), Contextos, Representaciones y la idea de Probabilidad Condicional, Investigaciones en Matemática Educativa, pp. 9-3, Grupo Editorial Iberoamérica, México. Puig, L., (996), Elementos de resolución de problemas. (Comares: Granada). Annex: Collection of problems for the tests. PROBLEMA : De todos los alumnos del instituto, un 3% practican baloncesto y fútbol y un 3% practican el baloncesto y no practican el fútbol. Sabemos que de los alumnos que no practican baloncesto un 4% hacen fútbol. Calcula la probabilidad de practicar fútbol. PROBLEMA : Un 3% de los huéspedes de un hotel practican el tenis y el golf y un 3% practican el tenis y no practican el golf. Además conocemos que de los huéspedes que no practican tenis un 4% practican golf. Calcula la probabilidad de que elegido un huésped al azar no practique ni tenis ni golf.problema 3: En una academia de idiomas un 3% de los alumnos estudian inglés y francés y un 3% estudian inglés y no estudian francés. Además, de los alumnos que no estudian inglés, un 4% estudian francés. Calcula la probabilidad de que estudie inglés elegido un alumno que estudia francés. PROBLEMA 9: En un instituto, la probabilidad de practicar baloncesto y fútbol es.3 y la probabilidad de practicar el baloncesto y no practicar el fútbol es 3. Sabemos que la probabilidad de que elegido un alumno de los que no practica baloncesto, éste practique fútbol es 4. Calcula la probabilidad de practicar fútbol. PROBLEMA : En un hotel, la probabilidad de que elegido un huésped al azar éste practique el tenis y el golf es 3 y la probabilidad de que practique el tenis y no practique el golf es 3. Además conocemos que la probabilidad de que elegido un huésped de los que no practican tenis éste practique golf es 4. Calcula la probabilidad de que elegido un huésped al azar no practique ni tenis ni golf. PROBLEMA : En una academia de idiomas, elegido un estudiante al azar la probabilidad de que estudie inglés y francés es 3 y de que estudie inglés y no estudie francés es 3. Además, elegido un alumno de los que no estudian inglés, la probabilidad de que estudie francés es de 4. Calcula la probabilidad de que estudie inglés elegido un alumno que
11 estudian francés. Además, de los alumnos que no estudian inglés, un 4% estudian francés. Calcula la probabilidad de que estudie inglés elegido un alumno que estudia francés. PROBLEMA 4: En una empresa el 55% de los trabajadores son mujeres. De las mujeres, el % se dedican a las tareas administrativas, y de todos los trabajadores, el 5% son hombres y administrativos. Calcula la probabilidad de ser mujer y no realizar tareas administrativas PROBLEMA 5: En una universidad el 55% de los estudiantes son mujeres. De éstas, el % estudian carreras de letras, y de todos los estudiantes, el 5% son hombres y estudian carreras de letras. Calcula la probabilidad de que elegido un estudiante al azar (hombre o mujer) estudie carrera de letras PROBLEMA 6: En un campamento de verano el 55% de los integrantes son niñas. De las niñas, el % realizan actividades acuáticas, y de todos los integrantes, el 5% son niños y realizan actividades acuáticas. Calcula la probabilidad de que eligiendo un integrante que realiza actividades acuáticas, éste sea niña. PROBLEMA 7: (Grupo Erema: M.A. Martín, J.M. Rey, M. Reyes, Estadística y Probabilidad, Bachillerato, Cuaderno 4, Grupo Editorial Bruño, Madrid, página 6, problema, cambiado y preparado para la prueba) Un 6% de los alumnos de un colegio aprobaron filosofía y un 7% matemáticas. Además, un 8% de los alumnos que aprobaron matemáticas, aprobaron también filosofía. Si Juan aprobó filosofía, qué probabilidad tiene de haber aprobado también matemáticas? PROBLEMA 8: Santos, D., (988), Matemáticas COU, Opciones C y D, Madrid: Santillana. p.48, problema, cambiado y preparado para la prueba) En un curso el porcentaje de aprobados en Historia (A) es 6 %. Para Matemáticas (B) es del 55 %. Sabiendo que p(b/a) = 7 %, cuál es la probabilidad de que, escogido al azar un alumno, resulte no haber aprobado ninguna de las dos asignaturas? es 3 y de que estudie inglés y no estudie francés es 3. Además, elegido un alumno de los que no estudian inglés, la probabilidad de que estudie francés es de 4. Calcula la probabilidad de que estudie inglés elegido un alumno que estudia francés. PROBLEMA : De los trabajadores de una empresa, la probabilidad de ser mujer es de 55. De las mujeres, la probabilidad de dedicarse a las tareas administrativas es de, y elegido un trabajador al azar, la probabilidad de ser hombre y administrativo es 5. Calcula la probabilidad de ser mujer y no realizar tareas administrativas. PROBLEMA 3: En una universidad, elegido un estudiante al azar, la probabilidad de que sea mujer es 55. De éstas, la probabilidad de que estudien carreras de letras es de, y elegido un estudiante al azar, la probabilidad de ser hombre y estudiar carrera de letras es de 5. Calcula la probabilidad de que elegido un estudiante al azar (hombre o mujer) estudie carrera de letras. PROBLEMA 4: La probabilidad de que los integrantes de un campamento de verano sean niñas es de 55. De las niñas, la probabilidad de realizar actividades acuáticas es de, y elegido un integrante al azar, la probabilidad de ser niño y realizar actividades acuáticas es de 5. Calcula la probabilidad de que eligiendo un integrante que realiza actividades acuáticas, éste sea niña. PROBLEMA 5: En un colegio, la probabilidad de aprobar filosofía es de 6 y la de aprobar matemáticas es de 7. Además, elegido un alumno de los que aprobaron matemáticas, la probabilidad de que aprobara filosofía es de 8. Si Juan aprobó filosofía, qué probabilidad tiene de haber aprobado también matemáticas? PROBLEMA 6: En un curso la probabilidad de aprobar Historia (A) es 6 y la de aprobar Matemáticas (B) es 5. Sabiendo que p(b/a) =.7, cuál es la probabilidad de que, escogido al azar un alumno, resulte no haber aprobado ninguna de las dos asignaturas?