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The assignment involves multiple probability and statistics problems, primarily focused on understanding sample spaces, calculating probabilities for different events, and proving properties related to probability spaces, including independence, conditional probability, and inclusion-exclusion principles. Students are asked to solve theoretical exercises involving probability distributions, events, and games of chance, as well as to demonstrate proofs based on probability axioms and definitions.

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The collection of problems presented for this assignment provides a comprehensive exploration of fundamental concepts in probability theory, challenging students to analyze theoretical scenarios and demonstrate formal understanding through calculations and proofs. These exercises are designed to deepen comprehension of probability distributions, set operations, independence, conditional probability, and combinatorial reasoning.

One of the fundamental tasks is to determine sample spaces and associated probabilities for simple random experiments. For example, the problem involving selecting a letter from the word "MISSISSIPPI" requires students to identify all possible outcomes and assign probabilities accordingly. Since each letter in "MISSISSIPPI" can be considered equally likely, the total number of letters is 11, with a particular count of each letter. The sample space includes all 11 outcomes, with specific outcome probabilities based on uniform selection or frequency. This kind of exercise emphasizes understanding uniform probability models and basic probability calculations.

Another essential topic covered is the probability of combined events, such as the probability of students smoking or drinking, considering overlaps. The problem involving students who smoke (25%), drink (60%), or do both (15%) demonstrates the application of the inclusion-exclusion principle, which helps compute the probability that a student either smokes or drinks. The formula states that P(smoke or drink) = P(smoke) + P(drink) - P(both). Substituting the given percentages reveals that 70% of students either smoke or drink, illustrating how these events overlap and how to account for that in probability calculations.

Proving properties of probability measures is also a key component. For instance, demonstrating that P(B) can be derived using the probabilities P(A), P(AC ∩ BC), and P(A ∩ B), given specific values, asks students to manipulate set operations and probability axioms to find unknown probabilities. Similarly, deriving maximum and minimum possible values for P(A ∩ B) with known P(A) and P(B) demonstrates the bounds of intersection probabilities, which are constrained by the inclusion-exclusion principle and the axioms of probability.

Set operations and the inclusion-exclusion principle are central to understanding the probability of unions of multiple events, as expressed in the formula involving three events A, B, and C. By proving that P(A ∪ B ∪ C) equals the sum of individual probabilities minus the intersections of pairs, plus the triple intersection, students reinforce their grasp of combinatorial probability and the properties of measures on sigma-algebras.

Game theory and probabilistic modeling of games of chance are also included. For example, calculating the probability that Alice wins a coin-flip game when she flips first involves understanding geometric probability distributions and the memorylessness property of fair coin flips. Similarly, analyzing an unfair coin with biased probabilities involves conditional probabilities on sequences of flips, with practical calculations determining probabilities of specific outcomes within a fixed number of flips.

Conditional probability, independence, and their proofs are vital topics, especially demonstrating that the complement of an event B intersected with A, conditioned on B, equals 1 minus the conditional probability of A given B. This reinforces foundational principles regarding how conditioning affects probabilities and how the axioms underpin these relations. Additionally, the independence of events A and B implies independence of their complements, illustrating the symmetric nature of independence.

The last set of problems involves analyzing joint distributions of die rolls and events related to parity, such as whether the first roll is odd or the second is even, and combined events like both rolls being even or both odd. These problems highlight the concepts of pairwise independence, mutual independence, and the importance of understanding how events relate logically and probabilistically within the framework of joint distributions.

Overall, this assignment aims to develop students' analytical skills in probability, emphasizing both computational proficiency and the ability to prove fundamental properties from axioms and definitions. Mastery of these problems prepares students to handle more complex stochastic models and reasoning tasks in advanced probability theory and statistics.

References

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• Devore, J. L. (2014). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
• Klenke, A. (2008). Probability Theory: A Comprehensive Course (2nd ed.). Springer.
• Kingman, J. F. C., & Taylor, S. (2013). Introduction to Probability. Cambridge University Press.
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• Ghahramani, Z. (2015). Probabilistic Machine Learning and Artificial Intelligence: Perspectives and Opportunities. Nature.