# The Blog

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## (3) Alan Cooper’s answer to Question: A bar of length ℓ is broken into three pieces at two random spots. What is the probability that the length of at least one piece is less than ℓ/20? Can anyone hlp me? Thks a lot – Quora

The question is ill-posed because there are many different ways of choosing two spots “at random”. BUT if we assume that the cuts are made independently with each chosen according to a uniform probability per unit length then the answer is the fraction of the big square that is not shaded in the diagram below…

## (3) Alan Cooper’s answer to Alice and Bob flip a biased coin, best $n$ out of $2n-1$ win. If the probability of Alice winning a flip is $p$, what is her chance of winning the series? – Quora

Since this is tagged with “Puzzles and Trick Questions” it may be that I am missing something. But my answer would be $\Sigma_{m=0}^{n-1}p^{2n-1-m}(1-p)^m$ . This follows the pattern of the best 2 out of 3 case where Alice has to win either two or three games – which happens in cases lww,wlw,wwl or www with…

## (3) Alan Cooper’s answer to How do I find the period of $e^{ix}$ without using trigonometry? – Quora

This question has been around for a while and has some decent answers. But I want to suggest a simpler and more intuitive version. (And it will be easier to follow if I replace the variable $x$ by $t$ so as not to confuse it with the real part of the complex function value.) First,…

## Awareness of Number

The question of whether, and if so how, our brains are Wired for Numbers has obvious implications for the teaching of mathematics, and also, I think, for the question of how we interpret the “reality” of number and quantity in a “philosophy of math” sense. Source: Is Your Brain Wired for Numbers? | The Scientist…

## Hello world!

Welcome to Alan’s Math Notes. This is where I am planning to restore and make available various on-line notes and learning resources that I either developed myself and/or found useful and freely available from other sources.