4.3. In a certain college town, 27.5 percent of the students failed Mathematics (Math), 19.8 percent failed Biology (Bio), and
11 percent failed both Math and Bio. A student is selected at random. Let A represent the event of failing Math, so that A' is
the probability of passing Math. Let B be the event of failing Bio and B' the event of passing Bio.
(j) If the student failed Math or Bio, what is the probability that he or she failed Math and Bio?
(k) If the student failed Math and Bio, what is the probability that he or she failed Math or Bio? Enter the solution.
(l) If the student passed Math or Bio, what is the probability that the student failed Math and Bio? Enter the solution.
To find the answers to these questions, we need to use some basic principles of probability and set theory. We'll break down each question and explain how to arrive at the answers.
(j) If the student failed Math or Bio, what is the probability that he or she failed Math and Bio?
To find the probability that the student failed Math and Bio (A ∩ B), we can use the formula: P(A ∩ B) = P(A) + P(B) - P(AUB), where P(AUB) represents the probability of the union of events A and B.
Given in the question:
P(A) = 27.5% (probability of failing Math)
P(B) = 19.8% (probability of failing Bio)
P(A ∩ B) = 11% (probability of failing both Math and Bio)
So, P(AUB) = P(A) + P(B) - P(A ∩ B)
= 27.5% + 19.8% - 11%
= 36.3%
Therefore, the probability that the student failed Math and Bio, given that the student failed Math or Bio, is 11% / 36.3%, which is approximately 0.303 (or 30.3%).
(k) If the student failed Math and Bio, what is the probability that he or she failed Math or Bio?
To find the probability that the student failed Math or Bio (AUB), given that the student failed both Math and Bio (A ∩ B), we can use the formula: P(AUB) = P(A) + P(B) - P(A ∩ B), as explained in the previous question.
Given in the question:
P(A) = 27.5% (probability of failing Math)
P(B) = 19.8% (probability of failing Bio)
P(A ∩ B) = 11% (probability of failing both Math and Bio)
So, P(AUB) = P(A) + P(B) - P(A ∩ B)
= 27.5% + 19.8% - 11%
= 36.3%
Therefore, the probability that the student failed Math or Bio, given that the student failed both Math and Bio, is 36.3% / 11%, which is approximately 3.3.
(l) If the student passed Math or Bio, what is the probability that the student failed Math and Bio?
To find the probability that the student failed Math and Bio (A ∩ B), given that the student passed Math or Bio (A' U B'), we need to subtract the probability of passing Math or Bio from 1 (since A U B represents the complement of A' U B').
Given in the question:
P(A) = 27.5% (probability of failing Math)
P(B) = 19.8% (probability of failing Bio)
P(A ∩ B) = 11% (probability of failing both Math and Bio)
Probability of passing Math (A') = 100% - P(A) = 100% - 27.5% = 72.5%
Probability of passing Bio (B') = 100% - P(B) = 100% - 19.8% = 80.2%
P(A' U B') = P(A') + P(B') - P(A' ∩ B')
= 72.5% + 80.2% - (100% - P(A ∩ B))
= 152.7% - 100% + P(A ∩ B)
= 52.7% + 11%
= 63.7%
Therefore, the probability that the student failed Math and Bio, given that the student passed Math or Bio, is 11% / 63.7%, which is approximately 0.173 (or 17.3%).