This article is a summary of a YouTube video "Why “probability of 0” does not mean “impossible” | Probabilities of probabilities, part 2" by 3Blue1Brown

Understanding Probabilities in Continuous Settings

TLDRWhen dealing with probabilities in continuous settings, we can't assign probabilities to individual values, but instead use probability density functions (PDFs) to describe the distribution. The area under the PDF represents the probability of a range of values. This approach solves the paradox of assigning probabilities to each individual value in an infinite set. The rules for combining probabilities of different sets are different in continuous settings compared to discrete settings. Measure theory helps to unite the two settings and provide a rigorous framework for probability calculations.

Key insights

🔢In continuous settings, probabilities cannot be assigned to individual values, but instead, we use probability density functions (PDFs) to describe the distribution.

🌡️The area under the PDF represents the probability of a range of values, solving the paradox of assigning probabilities to each individual value in an infinite set.

📚Measure theory provides a formal foundation for probability and helps unite the rules for combining probabilities in discrete and continuous settings.

⚖️Probability density functions (PDFs) allow us to calculate the probability of a range of values and answer questions about the probability of specific intervals.

🎯Probability density is better tied to possibility than probability, and different rules apply to combining probabilities of different sets in continuous settings.

Q&A

Why can't we assign probabilities to individual values in continuous settings?

In continuous settings, there are infinitely many possible values, and assigning a non-zero probability to each value would result in an infinite sum, which is not mathematically meaningful. Instead, we use probability density functions (PDFs) to describe the probability distribution over ranges of values.

How do probability density functions (PDFs) work?

A probability density function (PDF) assigns probabilities to ranges of values in a continuous distribution. The area under the PDF represents the probability of that range. The height of the PDF represents the probability density, which is a measure of probability per unit in the x-direction.

What is measure theory?

Measure theory is a branch of mathematics that provides a formal foundation for probability and helps to unify the rules for combining probabilities in discrete and continuous settings. It allows us to define probabilities for sets of values in a consistent and rigorous manner.

How are probability density functions (PDFs) used in practice?

Probability density functions (PDFs) are used to calculate the probability of a range of values and answer questions about the probability of specific intervals. They are widely used in statistics, machine learning, and other fields where continuous variables are encountered.

Why is probability density better tied to possibility than probability?

In continuous settings, the probability of any specific value is zero because there are infinitely many possible values. However, the possibility of an outcome is better represented by the probability density, which measures the likelihood per unit in the x-direction.

Timestamped Summary

00:00In continuous settings, probabilities cannot be assigned to individual values. Instead, we use probability density functions (PDFs) to describe the distribution.

05:38The area under the PDF represents the probability of a range of values, solving the paradox of assigning probabilities to each individual value in an infinite set.

06:50Measure theory provides a formal foundation for probability and helps unite the rules for combining probabilities in discrete and continuous settings.

07:51Probability density functions (PDFs) allow us to calculate the probability of a range of values and answer questions about the probability of specific intervals.

08:53Probability density is better tied to possibility than probability, and different rules apply to combining probabilities of different sets in continuous settings.