Lecture 3: Birthday Problem, Properties of Probability | Statistics 110

Harvard University・2 minutes read

The Birthday Problem explores the probability of two people sharing a birthday, with 23 individuals needed for a 50.7% chance of a match, leading to practical applications in computer science. The concept of inclusion-exclusion is discussed, with a formula involving adding individual probabilities, subtracting intersections, and adjusting for triple intersections, showcasing its practical application in probability scenarios like de Montmort's problem.

Insights

  • The Birthday Problem reveals that with just 23 individuals, there is a 50.7% chance of two people sharing the same birthday, showcasing the surprising nature of probability in group settings.
  • The concept of inclusion-exclusion in probability calculations is crucial, as seen in the de Montmort's problem example, where alternating sums are used to determine probabilities accurately, demonstrating the practical applications of this mathematical principle.

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Recent questions

  • What is the Birthday Problem?

    The Birthday Problem analyzes the likelihood of shared birthdays.

  • How can the probability of a birthday match be calculated?

    Calculate the probability of no match and subtract from 1.

  • What is the significance of "K choose 2" in the Birthday Problem?

    "K choose 2" pairs are crucial for understanding probability.

  • How can the probability of coincidences be explained in the Birthday Problem?

    Coincidences are inevitable due to numerous possibilities.

  • What practical applications does the Birthday Problem have?

    It highlights the importance of efficient data storage.

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Summary

00:00

"The Birthday Problem: Probability and Surprising Results"

  • The Birthday Problem is a fundamental probability problem that examines the likelihood of two people sharing the same birthday in a group.
  • The problem aims to determine the minimum number of people needed for a 50/50 chance of a birthday match.
  • Assumptions include excluding February 29th, considering 365 days in a year, and assuming equal likelihood of birthdays.
  • Independence among birthdays is assumed, meaning one person's birthday does not affect another's.
  • For groups larger than 365 people, the probability of a birthday match is 1 due to the Pigeon Hole Principle.
  • For groups less than or equal to 365 people, the surprising answer is 23 individuals for a 50.7% chance of a match.
  • Calculating the probability of no match first and then subtracting from 1 simplifies the computation.
  • With 50 people, the probability of a match rises to 97%, and with 100 people, it exceeds 99.999%.
  • The concept of "K choose 2" pairs is more relevant than the total number of people in understanding the probability of a birthday match.
  • The Birthday Problem has practical applications in computer science and highlights the importance of efficient data storage to avoid collisions.

14:10

"Calculating Probabilities and Understanding Coincidences"

  • With 23 people, there are 253 pairs of people, calculated as 23 choose 2, which is 253.
  • To easily multiply by 11, add the digits of the number and place the result in the middle, e.g., 23 x 11 = 253.
  • The concept of coincidences is explored, with 253 pairs of people potentially sharing the same birthday out of 365 days.
  • If two people have birthdays one day apart, the number of people needed for a 50% chance of this is 14.
  • The complexity of calculating probabilities for birthdays being the same or one day apart is acknowledged.
  • The idea that coincidences are inevitable due to the vast number of possible coincidences is discussed.
  • A recommendation is made to use probability applets to better understand and simulate the birthday problem.
  • The non-naive definition of probability is introduced, with two axioms: probability of empty set is 0, and full sample space is 1.
  • The axiom that the probability of a union of disjoint events is the sum of their individual probabilities is explained.
  • Properties of probability are discussed, including the probability of a complement being 1 minus the probability of the event, and the probability of a subset being less than or equal to the probability of the superset.

28:59

"Disjointification and Inclusion-Exclusion in Probability"

  • Probability of a union is often sought when sets are not disjoint, leading to the need to add probabilities and subtract intersections.
  • To prove this concept, a strategy involves making sets disjoint to apply axioms, leading to the term "disjointification."
  • Disjointification involves taking a union and rewriting it in a disjoint manner by excluding elements already present in one set.
  • Applying axiom two after disjointification results in the equation P(A) + P(B intersect A complement).
  • Wishful thinking is used to propose the equation P(A) + P(B) - P(A intersect B) for comparison.
  • The equation is proven true by showing that A intersect B and A complement intersect B are disjoint, leading to the conclusion that their union is B.
  • The concept discussed is a simple case of inclusion and exclusion, with a more general formula provided for multiple events.
  • The general inclusion-exclusion formula involves adding individual probabilities, subtracting intersections, and adjusting for triple intersections.
  • The formula extends to cases with more than two events, with a pattern of alternating inclusion and exclusion.
  • An example of applying inclusion-exclusion to de Montmort's problem, a card game probability scenario, showcases the practical application of the concept.

44:13

Calculating Card Probability with Factorials

  • The probability of finding the ace of spades in a deck of 52 cards is 1 in 52, regardless of its position, calculated as 1 over n or using n factorial.
  • When calculating the probability of specific cards appearing in certain positions in a deck, the formula involves factorials and simplifies to 1 over n(n-1) due to cancellations.
  • By applying inclusion-exclusion principles, the final probability calculation results in an alternating sum that closely resembles the Taylor series for e to the x, approximating to 1 minus 1 over E.
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