Question

III. Para el exámen de Estadística y probabilidad, un alumno sólo ha estudiado 13 de los 27 temas correspondientes a la materia del mismo. Éste se realiza extrayendo al azar dos temas y dejando que el alumno escoja uno de los dos para ser examinado del mismo. Hallar la probabilidad de que el alumno pueda elegir en el examen uno de los temas estudiados.

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Answer 1

The probability that the student can choose one of the topics studied in the exam is approximately 0.222 or 22.2.

Let's break down the problem:

The student has studied 13 out of 27 topics.

The student randomly extracts two topics.

The student chooses one of the two topics to be examined.

Now, let's find the probability that the student can choose one of the topics studied in the exam.

The total number of ways the student can extract two topics out of 27 is given by the combination formula:

C(n,r)= n!/ r!(n−r)!

In this case, n=27 (total topics) and r=2 (number of topics to be extracted).

C(27,2)= 27!/2!(27−2)!

C(27,2)= 27!/2!⋅25!

C(27,2)= 27⋅26/2

C(27,2)=27⋅13

C(27,2)=351

So, there are 351 ways to extract two topics.

Now, the student has only studied 13 topics out of the 27. The probability that both extracted topics are studied is given by the combination of choosing 2 topics out of the 13 studied ones:

C(13,2)= 13!/ 2!(13−2)!

C(13,2)= 13!/ 2!⋅11!

C(13,2)= 13⋅12/2

C(13,2)=78

So, there are 78 ways to extract two studied topics.

Now, the probability that the student can choose one of the topics studied is the ratio of the number of ways to choose two studied topics to the total number of ways to choose any two topics:

P(studied)= Number of ways to choose 2 studied topics/ Total number of ways to choose 2 topics

P(studied)= 78/351

P(studied)≈0.222

Therefore, the probability that the student can choose one of the topics studied in the exam is approximately 0.222 or 22.2.

Question

III. For the Statistics and Probability exam, a student has only studied 13 of the 27 topics corresponding to the subject. This is done by randomly extracting two topics and letting the student choose one of the two to be examined from it. Find the probability that the student can choose one of the topics studied in the exam.