# Probability

## Bayes and Virus Testing

News about testing for viruses has reminded me of a couple problems that I linked to some time ago, but never dealt with directly. The question is, given data such as the result of a (fallible) blood or swab test, how sure can we be of the results? The answer is sometimes surprising. False positives …

## Combinatorics and Coefficients

(A new question of the week) A question from last August gave us some nice problems reminiscent of the Binomial Theorem, which were very deserving of discussion. Three problems The question came from Arsh: I have some coefficient problems which I am unable to solve. I don’t know if a single concept will work for …

## Rank of a Binary Number

(A new question of the week) A few months ago, I wrote about Ranking a Word Among Its Permutations, that is, finding where a word would be found in an ordered list of all possible “words” made by permuting its letters. The problem in general requires a (sometimes lengthy) algorithm. A month or so later, …

Last time, we looked at the basic definition of independent events. This time I want to explore some deeper questions about the concept. Independence by the numbers We’ve seen that, informally, we think of independent events as not affecting one another’s probabilities. Mathematically, though, independence is defined by the fact (which is implied by that …

## What Are Independent Events?

The concept of independent events can be both very simple and easily misunderstood. We’ll be looking at several explanations of the idea, starting with the basics and then digging into some deeper questions that are often overlooked. What is independence? We can start with this question from 1998, asking for the basics: Independent and Dependent …

## Ranking a Word Among Its Permutations

(A new question of the week) There are some topics that appear to be standard in certain parts of the world, but far less familiar in our own. Sometimes it takes two of us to recognize what a student is asking, due to language issues and different past experience with such questions. This is an …

## Interpreting and Solving a Counting Problem

(An archive question of the week) Combinatorics can be inherently tricky; making up your own problem is doubly so. Here we have a problem created by a teacher, who then is not entirely sure what it means. How can we figure out what meaning to give it? Combine that with working out how to solve …

## Arranging Letters with Duplicates

(A new question of the week) Here is a recent discussion with a frequent user of our service, Kurisada, about combinatorics. He is new to the subject, so this involved several introductions to new ideas. Arranging 8 letters, 2 identical There are 8 letters : A, B, H, N, O, S, U, U I was …

## How Many Different Pizzas?

(An archive question of the week) We’ve been looking at examples of extended discussions with students about various kinds of problems. Here, we have one (not from a student) that led to some good thinking about combinatorics – the techniques of counting the ways something can happen. The problem: Triple toppings Here’s the question, from …

## More on Gender Probability: Twins

(An archive question of the week) Recently we looked at the question of how likely a two-child family with a boy is to have another boy (or, to the contrary, to also have a girl). Searching for those questions turned up another one of interest involving the gender of a pair of siblings: How do …