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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.