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Lecture 2: Story Proofs, Axioms of Probability | Statistics 110

Harvard University・33 minutes read

Common sense is crucial for approaching homework and exams, using calculators for homework is optional, but not allowed in exams. Probability problems require assuming equally likely outcomes, understanding patterns and structures is essential for complex counting scenarios.

Insights

  • Utilizing common sense in approaching homework and exams, even when encountering counterintuitive results, is crucial for success in mathematics.
  • Understanding the nuances of probability and combinatorics involves careful consideration of equally likely outcomes, distinguishing between scenarios where order matters and where it doesn't, and employing strategic practices like story proofs to enhance comprehension and problem-solving skills.

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Recent questions

  • How should one approach homework and exams?

    By applying common sense and checking answers thoroughly.

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Summary

00:00

Effective Homework and Exam Strategies

  • Common sense is crucial in approaching homework and exams, even when faced with counterintuitive results.
  • Calculators can be used for homework but are not necessary; exams do not allow calculators.
  • Leaving expressions like "52 choose 5" is acceptable and even beneficial for self-annotation.
  • Simplifying expressions like "4 divided by 2 times 1" to "2" is preferred.
  • Checking answers involves more than just reviewing; it includes trying special cases and alternative methods.
  • Labeling individuals or objects in problems is highly useful for clarity and understanding.
  • In probability problems, assuming equally likely outcomes is essential for applying the naive definition.
  • A key distinction exists between scenarios where order matters and where it doesn't in team-splitting problems.
  • The naive definition of probability is sufficient for homework questions but requires careful framing of equally likely outcomes.
  • The formula for picking k times from n objects with replacement and order indifference is n plus k minus 1 choose k ways.

14:21

Counting Principles in Combinatorics: A Primer

  • 0 factorial is 1, and n choose 0 is 1 for any n, as choosing none out of a group of n people has one way.
  • Changing k to n in the formula n minus 1 choose n results in 0, as picking n people out of n minus 1 is impossible.
  • When k equals 1, n choose 1 equals n, indicating that picking once results in n, regardless of replacement or order.
  • For n equals 2, the formula k plus 1 choose k simplifies to k plus 1, showcasing the simplest nontrivial example.
  • The problem can be viewed as putting k indistinguishable particles into n distinguishable boxes, leading to the formula n plus k minus 1 choose k.
  • Drawing a diagram with dots and separators helps visualize the problem, with n plus k minus 1 positions to choose from.
  • The proof of the formula's correctness lies in selecting positions for dots and separators, resulting in n plus k minus 1 choose k.
  • In the n equals 2 case, flipping two coins can illustrate the concept, with the possibility of three equally likely outcomes.
  • A historical example involving Bose and Einstein showcases the significance of similar counting principles in predicting the Bose-Einstein condensate.
  • Understanding patterns and structures, as well as recognizing equivalent problems, is crucial in solving complex counting scenarios.

29:10

"Story Proofs Reveal Mathematical Insights"

  • 70 years later, a prediction was empirically observed, showcasing bizarre properties.
  • Viewing coins as labeled is useful for counting and physics, but caution is advised with naive probability definitions.
  • Story proofs offer interpretations rather than algebraic proofs, exemplified by 10 choose 4 equals 10 choose 6.
  • The strategic practice includes interpreting mathematical proofs through story proofs.
  • N choose k equals n choose n minus k is a valuable fact, demonstrated through story interpretation.
  • The identity n minus 1 choose k minus 1 equals k times n choose k is derived through story proofs.
  • Vandermonde's identity, m plus n choose k, is proven through a story of selecting people from two groups.
  • The non-naive definition of probability involves a probability space with sample space S and function P.
  • P assigns probabilities between 0 and 1 to events, following axioms where the empty set has a probability of 0 and the full space has a probability of 1.
  • Impossible events have a probability of 0, while certain events have a probability of 1 in the non-naive definition of probability.

44:33

Probability Axioms: Universe and Countable Unions

  • The two axioms of probability are: the first one states that the probability of the whole universe is 1, and the second one explains that the probability of a countable union of non-overlapping events is equal to the sum of their individual probabilities.
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