2. A universal set consists of numbers from 1 to 20. Set A are the factors of 20 and set B are the multiples of 3. Show through the Venn diagram the sets of numbers. Answer the following questions: Selecting a number at random between 1 and 12,
(i) What is the probability of selecting a number that is a factor of 20?
(ii) What is the probability of selecting a number that is a factor of 20 and a multiple of 3?
(iii) What is the probability of selecting a number that is not a multiple of 3?
3. In the final exams, 40% of the students failed language subject, 25% failed in sciences, and 19% failed both language and physics. What is the probability that a randomly selected student failed in science given that he passed language?
(i) What is the probability of selecting a number that is a factor of 20?
(ii) What is the probability of selecting a number that is a factor of 20 and a multiple of 3?
(iii) What is the probability of selecting a number that is not a multiple of 3?
3. In the final exams, 40% of the students failed language subject, 25% failed in sciences, and 19% failed both language and physics. What is the probability that a randomly selected student failed in science given that he passed language?
Answers & Comments
Answer:
1. To create a Venn diagram for this problem, we can draw two intersecting circles to represent sets A and B. Set A consists of the factors of 20, which are 1, 2, 4, 5, 10, and 20. Set B consists of the multiples of 3, which are 3, 6, 9, 12, 15, and 18.
(i) To find the probability of selecting a number that is a factor of 20, we need to count the number of elements in set A and divide by the total number of elements in the universal set. There are 6 elements in set A and 20 elements in the universal set, so the probability is:
P(factor of 20) = 6/20 = 3/10
(ii) To find the probability of selecting a number that is a factor of 20 and a multiple of 3, we need to count the number of elements in the intersection of sets A and B (which is just 6) and divide by the total number of elements in the universal set. So the probability is:
P(factor of 20 and multiple of 3) = 6/20 = 3/10
(iii) To find the probability of selecting a number that is not a multiple of 3, we need to count the number of elements in set A that are not in set B (which are 1, 2, 4, and 20) and divide by the total number of elements in the universal set. So the probability is:
P(not a multiple of 3) = 4/20 = 1/5
3. Let L be the event that a student fails language, and let S be the event that a student fails science. We are given that P(L) = 0.4, P(S) = 0.25, and P(L and S) = 0.19.
We want to find P(S|L), the probability that a student fails science given that they passed language. This is a conditional probability, which we can find using Bayes' theorem:
P(S|L) = P(L and S) / P(L)
We are given that P(L and S) = 0.19 and P(L) = 0.4, so we can substitute these values into the formula:
P(S|L) = 0.19 / 0.4
P(S|L) = 0.475
Therefore, the probability that a randomly selected student failed in science given that he passed language is 0.475 or 47.5%