Question

let the universal set, u, be the set of students at central high, a be the set of students taking algebra, and c be the set of students taking chemistry. if a student is selected at random, describe in words what is meant by each of the following probabilities.

a. ( p(a cup c) )
b. ( p(a cap c) )
c. ( 1 - p(c) )

a. describe the meaning of the probability ( p(a cup c) ) in words. choose the correct answer below.
a. the probability of a student not taking chemistry
b. the probability of a student not taking algebra
c. the probability of a student taking algebra or chemistry
d. the probability of a student taking algebra and chemistry

b. describe the meaning of the probability ( p(a cap c) ) in words. choose the correct answer below.
a. the probability of a student not taking algebra
b. the probability of a student taking algebra or chemistry
c. the probability of a student not taking chemistry
d. the probability of a student taking algebra and chemistry

Explanation:

Part a

To determine the meaning of \( P(A \cup C) \), we recall the definition of the union of two sets. The union of sets \( A \) and \( C \), denoted \( A \cup C \), consists of all elements that are in \( A \), in \( C \), or in both. In the context of probability, \( P(A \cup C) \) represents the probability that a student is in set \( A \) (taking algebra), set \( C \) (taking chemistry), or both. Now let's analyze the options:

• Option A: "The probability of a student not taking chemistry" refers to \( P(\overline{C}) \), not \( P(A \cup C) \).
• Option B: "The probability of a student not taking algebra" refers to \( P(\overline{A}) \), not \( P(A \cup C) \).
• Option C: "The probability of a student taking algebra or chemistry" matches the definition of \( P(A \cup C) \) since the union corresponds to "or" (inclusive or, meaning in \( A \), in \( C \), or in both).
• Option D: "The probability of a student taking algebra and chemistry" refers to \( P(A \cap C) \), not \( P(A \cup C) \).

To determine the meaning of \( P(A \cap C) \), we recall the definition of the intersection of two sets. The intersection of sets \( A \) and \( C \), denoted \( A \cap C \), consists of all elements that are in both \( A \) and \( C \). In the context of probability, \( P(A \cap C) \) represents the probability that a student is in both set \( A \) (taking algebra) and set \( C \) (taking chemistry). Now let's analyze the options:

• Option A: "The probability of a student not taking algebra" refers to \( P(\overline{A}) \), not \( P(A \cap C) \).
• Option B: "The probability of a student taking algebra or chemistry" refers to \( P(A \cup C) \), not \( P(A \cap C) \).
• Option C: "The probability of a student not taking chemistry" refers to \( P(\overline{C}) \), not \( P(A \cap C) \).
• Option D: "The probability of a student taking algebra and chemistry" matches the definition of \( P(A \cap C) \) since the intersection corresponds to "and" (being in both sets).

Answer:

C. The probability of a student taking algebra or chemistry

Part b