Two novel proofs show that the sum of a specific pair of normal random variables is not normal are established in this note. This is one of the most often misunderstood facts by first-year students in probability theory and statistics. The first proof is concise using the moment generating function. The second proof checks whether the moments of…

Take a family of independent events. If some of these events, or all of them, are replaced by their complements, then independence still holds. This fact, which is agreed upon by the members of the statistical/probability communities, is tremendously well known, is fairly intuitive and has always been frequently used for easing probability…

In this paper, we give possible suggestions for a classroom lesson about an application of probability using basic mathematical notions. We will approach to some combinatoric results without using "induction", "polynomial identities" nor "generating functions", and will give a proof of the "Vandermonde…

This article shows a generalization of Galileo's "passe-dix" game. The game was born following one of Galileo's [G. Galileo, "Sopra le Scoperte dei Dadi" (Galileo, Opere, Firenze, Barbera, Vol. 8). Translated by E.H. Thorne, 1898, pp. 591-594] explanations on a paradox that occurred in the experiment of tossing three fair "six-sided" dice.…

This article sets out some of the rationale and arguments for making major changes to the teaching and learning of statistical inference in introductory courses at our universities by changing from a norm-based, mathematical approach to more conceptually accessible computer-based approaches. The core problem of the inferential argument with its…

This note presents an alternative approach to the reasoning process and derivation of the hypergeometric probability mass function (pmf), and contrasts it with a binomial model. It utilizes the essential concept of sampling without replacement directly in the development of the mass function.

In the paper we present a simple game that students can play in the classroom. The game can be used to show that random variables can behave in an unexpected way: the expected mean can tend to zero or to infinity; the variance can tend to zero or to infinity. The game can also be used to introduce the lognormal distribution. (Contains 1 table and…

In the simple one-dimensional random walk setup, a path is described as follows. Toss a coin. If the result is head, score +1 and move one step forward; otherwise score -1 and move one step backward. One is interested to know the position after a given number of steps. In this paper, once again a coin-tossing experiment is carried out. But this…

Although errors in reasoning about conditional probability have been the focus of interest of psychologists for a long time, the development of conditional reasoning in school students has received little attention. This paper considers the responses of 69 students across grades 3 to 13 in an attempt to model the development of appropriate…

Reports on a study designed to provide baseline data in the area of conjunction and conditional events. Cross-sectional and longitudinal analyses revealed improvement with grade in expressing probability numerically and distinguishing conditional events, but no change in incidence of conjunction errors. (Author/MM)

Since the game SET[R] was first introduced to the public in 1993, it has stimulated some interesting studies. While the game itself is rather straightforward, a plethora of decent mathematical questions lies beneath the surface. It is perhaps because the game ties in so closely with such an underlying mathematical term that its implications can be…

The traditional approach to expressing cumulants in terms of moments is by expansion of the cumulant generating function which is represented as an embedded power series of the moments. The moments are then obtained in terms of cumulants through successive reverse substitutions. In this note we demonstrate how cumulant-moment relations are…

In this note we consider an experiment involving an urn and k balls with numbers 1, 2, 3, ..., k. The experiment consists of drawing n balls either with replacement or without replacement. We note some surprising results.

In this issue's "Classroom Notes" section, the following papers are discussed: (1) "Constructing a line segment whose length is equal to the measure of a given angle" (W. Jacob and T. J. Osler); (2) "Generating functions for the powers of Fibonacci sequences" (D. Terrana and H. Chen); (3) "Evaluation of mean and variance integrals without…