High School

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**Problem 1:** Let A, B, and C be independent events associated with a random experiment modeled with a probability measure P.

- Is \( A \cap B \) independent of C?
- What about \( A \cup B \)?

Justify your answers with a mathematical argument or counterexample.

**Problem 2:** To practice for a Colorado Spelling Bee Contest, Arapahoe Ridge High (A) and Boulder High (B) have selected their best five spellers to compete at a weekend mock contest.

- Let H denote the highest rank to be obtained by an A-student. Determine the probability mass function of H assuming that each of the possible \(10!\) rankings are equally likely.

**Note:** Identify 1 with the highest rank and 10 with the lowest.

Answer :

Final answer:

Problem 1: The intersection and union of independent events may or may not be independent of another event. Problem 2: The probability mass function of the highest rank attained by an A-student can be determined by considering all possible rankings.

Explanation:

Problem 1: In general, the intersection (Aâ‹‚B) of two independent events A and B is not necessarily independent of another event C. The same applies to the union (AUB) of two independent events. Independence in probability theory depends on the specific events and the underlying probability measure. A counterexample can be constructed to show that Aâ‹‚B and AUB can be dependent on C.

Problem 2: In this problem, the highest rank H attained by an A-student can be any number from 1 to 5 (since there are 5 students). The probability mass function of H can be determined by considering all possible combinations of rankings for the A-students. Each rank from 1 to 5 has a probability of 1/5. Therefore, the probability mass function of H is:

P(H=h) = 1/5 for h = 1, 2, 3, 4, 5.

Learn more about Probability Theory here:

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