A Polynomial Inequality: Exploration vs Proof
(A new question of the week) We have had a number of challenging questions about inequalities recently. I want to show one of those here, because it involved a useful discussion about how to prove them.
Three Meanings of “Percentile”
(An archive problem of the week) Having just discussed quartiles, I want to look at related issues concerning percentiles. There, I briefly mentioned different perspectives on the concept of quartile, and focused on differences in the details of the calculations; here I will focus mostly on the different perspectives, and then touch on variations in …
The Many Meanings of “Quartile”
Some time ago I discussed various issues pertaining to the concept of median in statistics. The same issues, and more, affect the concept of quartile (the median being the second quartile), so much so that different statistical software packages produce many different answers for quartiles. I have seen this affect students, who are taught one …
Conditional Probability and Multiple Choice
(A new question of the week) A recent question about probability has ties to Venn diagrams, tables, and Bayes’ Theorem. Questions about answering multiple-choice questions are common; this one offers a twist that provided opportunity to discuss several important concepts. Here is the initial question, from August: On a multiple choice question, only one answer is …
Rolling a 6 on Three Dice
(An archive question of the week) One of the discussions we looked at last time involved rolling three dice and getting at least one six. I didn’t go into detail on the calculation there; but I found another place where we discussed it at length. We’ll look at that here. A wrong way and a …
The Gambler’s Fallacy
Probability seems simple enough to many people that it can fool them into wrong conclusions. We have had many questions that involve the “Gambler’s Fallacy”, both from people who naively assume it without thinking, and from some who defend it using technical ideas like the Law of Large Numbers. The Gambler’s Fallacy Here is a …
Mixing Trig and Inverse Trig Functions
(A new question of the week) Since we’ve been doing a little trigonometry this week, let’s look at a recent set of questions from a student in the Philippines, all about compositions of trig and inverse trig functions. This is a topic with some interesting little surprises, and several of us answered, to give a …
A Tank with a Conical End
(An archive question of the week) Last time I surveyed what we have said about the volume of liquid in various kinds of tanks. One more special case I ran across deserved more detailed attention, because it demonstrates in detail how to do the calculations without much knowledge of calculus. The problem Here is the …
Measuring Water in a Tank
Over the years, we have received a huge number of questions asking about how to find the amount of liquid (water, oil, …) in a tank, usually a horizontal cylindrical tank. The simplest case involves a rather complicated formula; from there, we can reverse the formula (finding the depth for a given volume), or we …
Venn Diagrams: Language Issues
(A new question of the week) I mentioned that we have had a number of questions related to Venn diagrams recently. Here I would like to show a couple of these, from a Philippine student. Even fluent English speakers can get confused in these problems; observing how a student new to the language misinterprets details …