## Archive for February, 2023

### A Puzzle: Two-step Selection of a Digit

Here is a simple problem in probability.

(1) Pick a number k between 1 and 9. Assume all digits are equally likely.

(2) Pick a number m in the range from 1 to k.

What is the probability distribution for the number m?

A graph of the probability distribution is shown in the figure here.

Probability distribution for a decimal digit selected in a two-step process.

Can you derive a formula for this probability distribution?

Can you generalise it to the range from 1 to 10^n?

Can you relate this problem to Benford’s Law [described here]?

Solution, and more on Benford’s Law, next week.

### Weather Warnings in Glorious Technicolor

Severe weather affects us all and we need to know when to take action to protect ourselves and our property. We have become familiar with the colourful spectrum of warnings issued by Met Éireann.

For several years, Met Éireann has issued warnings of extreme weather. These depend on the severity of the meteorological event and the level of confidence in the forecast. They are formulated using forecasts produced by computer, algorithms that determine the likely impacts of extreme weather, and the expertise of the forecasters [TM242 or search for “thatsmaths” at irishtimes.com]. Continue reading ‘Weather Warnings in Glorious Technicolor’

### Ford Circles & Farey Series

Lester R Ford, Sr. (1886–1967).

American mathematician Lester Randolph Ford Sr. (1886–1967) was President of the Mathematical Association of America from 1947 to 1948 and editor of the American Mathematical Monthly during World War II. He is remembered today for the system of circles named in his honour.

For any rational number ${p/q}$ in reduced form (${p}$ and ${q}$ coprime), a Ford circle is a circle with center at ${(p/q,1/(2q^{2}))}$ and radius ${1/(2q^{2})}$. There is a Ford circle associated with every rational number. Every Ford circle is tangent to the horizontal axis and each two Ford circles are either tangent or disjoint from each other.

### From Wave Equations to Modern Telecoms

Mathematics has an amazing capacity to help us to understand the physical world. Just consider the profound implications of Einstein’s simple equation ${E = m c^2}$. Another example is the wave equation derived by Scottish mathematical physicist James Clerk Maxwell. Our modern world would not exist without the knowledge encapsulated in Maxwell’s equations. Continue reading ‘From Wave Equations to Modern Telecoms’