Lecture 3: Birthday Problem, Properties of Probability | Statistics 110 | Summary and Q&A

TL;DR

The Birthday Problem explores the probability of two or more people sharing the same birthday in a group, while the Inclusion-Exclusion Principle is a useful tool for calculating the probability of a union of events.

Key Insights

• 🥳 The Birthday Problem highlights the surprising probability of finding a birthday match in a group, even with a relatively small number of people.
• 👻 The Inclusion-Exclusion Principle is a powerful tool for calculating the probability of a union of events, allowing for complex calculations to be simplified.
• #️⃣ Symmetry can be used to reduce the number of calculations and simplify the application of the Inclusion-Exclusion Principle.

Transcript

So I wanted to do one of the most famous problems in probability to start with, and then we'll go back into the non-naive definition. At the very end of last time, we started the non-naive definition, so we'll get back to that in a little bit. But first, this is a problem that everyone who studies probability should be familiar with, that's called ... Read More

Questions & Answers

Q: What is the Birthday Problem?

The Birthday Problem examines the probability of two or more people sharing the same birthday in a group, considering factors such as the number of people in the group and assumptions about birthdays.

Q: Why is it surprising that only 23 people are needed for a 50/50 chance of a birthday match?

It is surprising because intuitively, most people think that a larger number of people would be needed to have a 50/50 chance of finding a birthday match. However, the calculations show that even with a small group, the probability is much higher than expected.

Q: What is the Inclusion-Exclusion Principle?

The Inclusion-Exclusion Principle is a method used in probability to calculate the probability of the union of events. It involves subtracting the intersections of events and adding back the double-counted overlaps to obtain the correct probability.

Q: How can the Inclusion-Exclusion Principle be applied to the matching problem?

In the matching problem, the Inclusion-Exclusion Principle can be used to calculate the probability of finding at least one card in a shuffled deck whose position matches the number on the card. By including and excluding different events, the probability can be determined.

Summary

This video discusses the Birthday Problem, which is the probability of finding at least one pair of people with the same birthday in a group of people. The video explains the problem, makes assumptions about birthdays, and presents the naive definition of probability. It then introduces the non-naive definition of probability, discussing the axioms that define it and deriving properties of probability using these axioms. Finally, the video explores the inclusion-exclusion principle and applies it to solve the matching problem in a card game.

Questions & Answers

Q: What is the Birthday Problem?

The Birthday Problem is the probability of finding at least one pair of people with the same birthday in a group.

Q: What assumptions are made about birthdays in the problem?

The assumptions made are that February 29th is excluded, there are 365 days in a year, and all other days are equally likely.

Q: How is independence defined in the problem?

Independence in this context means that one person's birthday has no effect on anyone else's birthday.

Q: What is the probability when the number of people is greater than 365?

If the number of people is greater than 365, the probability is 1 since there are more people than days in a year.

Q: How do you compute the probability of no match in the Birthday Problem?

The probability of no match is computed using the naive definition of probability, assuming equally likely birthdays. The numerator is the product of the number of possible birthdays for each person, while the denominator is 365 raised to the power of the number of people.

Q: What is the probability of a match in the Birthday Problem?

The probability of a match is 1 minus the probability of no match.

Q: How does the probability of a match change with different numbers of people?

As the number of people increases, the probability of a match also increases. For example, with 23 people, the probability is around 50.7%.

Q: What is the intuition behind the probability of a match increasing with a small number of people?

The intuition is that there are a large number of possible coincidences that could occur, and with a small number of people, the number of possible coincidences is relatively high compared to the number of days in a year.

Q: What are the two axioms of the non-naive definition of probability?

The two axioms are that the probability of the empty set is 0 and the probability of the full sample space is 1.

Q: What is the property of probability that states the probability of the complement of an event?

The property states that the probability of the complement of an event is 1 minus the probability of the event.

Q: How does the probability of an event change when it is contained in another event?

If one event is contained in another event, the probability of the contained event is less than or equal to the probability of the containing event.

Takeaways

The Birthday Problem is a fascinating probability problem that explores the likelihood of finding at least one pair of people with the same birthday in a group. The problem can be solved using the non-naive definition of probability, which is defined by two axioms. Properties of probability, such as the complement rule and containment rule, can be derived using these axioms. The inclusion-exclusion principle is a powerful tool for finding the probability of unions in more complex problems. The matching problem in a card game is an example that can be solved using inclusion-exclusion, showing the applicability of this technique in various scenarios.

Summary & Key Takeaways

• The Birthday Problem seeks to determine the likelihood of two or more people sharing the same birthday in a group.

• The probability is influenced by the number of people in the group and certain assumptions, such as excluding February 29th and assuming all other days are equally likely.

• Surprisingly, it only takes 23 people to have a 50/50 chance of a birthday match, and with 100 people, the probability is greater than 99.999%.

• The Inclusion-Exclusion Principle is a method used to calculate the probability of a union of events by subtracting the intersections and adding back the double-counted overlaps.