Wikipedia:Reference desk/Archives/Mathematics/2018 November 19

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November 19

Lottery prizes

Why is there such an extreme difference between the jackpot and the lowest prizes (three numbers)? How might one go about flattening the "pyramid" to give more people moderate winnings while reducing the extremely huge jackpots? Roger (Dodger67) (talk) 10:08, 19 November 2018 (UTC)

The article Lottery mathematics explains how to calculate the probabilites of each outcome for typical lotteries, but the reasons why lottery prizes are skewed the way they are not related to mathematics, but business. Lotteries with larger jackpots sell more tickets than lotteries where the prizes are a closer match to the probabilities, so lotteries reduce the value of lower prizes to make the larger prizes more attractive, and thus make more money from ticket sales. Iffy★Chat -- 10:42, 19 November 2018 (UTC)

Finding suitable substitutions for integration by substitution

Are there any general methods for finding helpful substitutions? By "general", I really mean applicable to wide classes of integrals that come up reasonably often. — Preceding unsigned comment added by Star trooper man (talk • contribs) 13:15, 19 November 2018 (UTC)

Not exactly, but when you see something like $\cos(x^{2}+1)$ in an integral somewhere, it's a pretty safe bet that a substitution of $u=x^{2}+1$ is going to be helpful. See also Trigonometric substitution (which needs some cleanup, but it's another wide class that comes up which you may or may not have seen so far) for another, less obvious type that can come up. –Deacon Vorbis (carbon • videos) 14:14, 19 November 2018 (UTC)

I did know about those.

However, I'm not trying to solve exam questions in particular (or rather, I am, but for the benefit of my students). I was hoping if there were any general rules that I could teach in guessing good substitutions.--Leon (talk) 16:21, 20 November 2018 (UTC)

You might want to take a look at Seán Stewart's How to Integrate It: A Practical Guide to Finding Elementary Integrals, Cambridge (2017). Several relevant chapters. In particular for trig integrals there is the Tangent half-angle substitution as a last resort, of course.John Z (talk) 16:58, 20 November 2018 (UTC)