Take a guided, problem-solving based approach to learning Probability. These compilations provide unique perspectives and applications you won't find anywhere else.
Sharpen your skills with these quizzes designed to check your understanding of the fundamentals.
If there are only a handful of objects, then you can count them with a moment's thought, but the techniques of combinatorics can extend to quickly and efficiently tabulating astronomical quantities.
Seek simple and succinct solutions in these systems by sussing-out the significant symmetries.
You are in the complement of the set of people who are not Brilliant.
As head of Brilliant's cybersecurity, you need to know how many different passwords can be created by rearranging the digits 1, 2, 3, 4, 5, and 6. Can you count them all?
If a coin is flipped twenty times, how many ways are there to get exactly ten tails and ten heads?
How many 5 block paths can take you home if walking 2 blocks north then 3 blocks west got you here? This should feel familiar if you're from NYC, Chicago, Barcelona, or Toronto!
When Cantor introduced his classification of multiple infinities, he was vehemently rejected by most mathematicians. Ye be warned: contemplating the continuum hypothesis can drive anyone a little mad!
Familiarize yourself with the foundational notation, tools and concepts for the operations that are applied to sets.
Count to 100. How many of those numbers are odd or multiples of 5? Since 50 are odd and 20 are multiples of 5, at first glance the answer is 70...
Bijections, surjections, and injections are three types of functions which associate the elements between two sets. For example, each word in this sentence can be mapped to exactly one in the last.
If you have 12 pigeons and there are only 11 roosts, then at least one roost will be quite cozy.
How can five candies be distributed among 3 friends? Well, the answer depends on if you can tell the candies apart!
Whether you're trying to analyze a political poll, the results of a scientific study, or just your sleeping habits, data is the best tool for understanding the world around you. Dive in and see how!
Data is only as good as how it is presented. How do you take hundreds or thousands of data points and create something a human can understand? Check out the world of charts, graphs, and more.
How often will a die come up "4"? How likely is it to rain tomorrow? Probability is one of the most powerful frameworks for modeling the world around us.
Conditional probability is the art of updating probabilities based on given information. What is the probability that the sidewalk is wet? And what if we know that it rained a few hours ago?
If you play roulette, are you going to break-even? Maybe sometimes, but not if you play forever, because the expected value is negative. The expected value finds the average outcome of random events.
Would you rather get paid $2 for flipping heads, or $6 for rolling a "1"? The expected value is the same ($1)...but the bets are different! Variance and standard deviation add color to probability.
Sometimes, probability questions can be interpreted geometrically, from simple examples like throwing darts to surprising applications like catching the bus!
How many heartbeats do you have each minute? How many points will your favorite team score in their game tonight? These any many other real-world values can be modeled by discrete random variables.
When will that bus finally arrive? How hot is it going to be outside today? These any many other real-world values can be modeled by continuous random variables.
What is the probability that the sum of two dice will be less than 5? How many throws will it take you to hit your first "bullseye" on a dart board? Use discrete probability distributions to find out!
How much variation should you have in your blood pressure? How likely is that stock price to double by the end of the year? Use continuous probability distributions to find out!
From weather conditions to baseball scores to stock performances, many probabilistic real-world systems can be modeled with Markov chains.
The Fibonacci sequence is not the only one that is defined recursively. Learn how to solve combinatorics problems with recursion, and how to turn recurrence relations into closed-form expressions.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... They show up in nature, they show up in math, and they've got some beautiful properties.
Take your recursion skills to the next level. If you've got a recurrence relation but no computer, how can you find a closed form? What about asymptotic behavior? How fast do rabbits reproduce?
If you have 12 pigeons and there are only 11 roosts, then at least one roost will be quite cozy.
Whether you’re a master of games or just playing around, learn how combinatorial ideas can be used to analyze and solve games such as Nim.
Learn about advanced problem-solving tactics such as Construction, the Extremal Principle, and the Invariant Principle, and you'll be crushing tricky problems in no time.
Any connected group of things can be represented as a graph: cities and roads, people and friendships, and more. Learn why an even number of people have an odd number of friends.
This combinatorial tool is used for solving recurrence relations and other difficult problems.
If coloring sounds pretty fun, it's because it is! See how coloring can turn a complex combinatorial problem into a one-line solution.
Browse through thousands of Probability wikis written by our community of experts.
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