Hamilton Lottery

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What are your chances of winning this lottery?

Context: Hamilton is a sold-out musical on Broadway. When a show is sold out, the theater will often do a lottery. They save 20 front-row seats at $10 apiece. We went to the Hamilton lottery and there were so many people….so few seats…. I wanted to calculate our chances of winning. I told the people running the lottery that I wanted to make a video for a math problem, and they were game. They even let me stand up on their platform and take pictures of their little counter-clicker device that kept track of how many people were in the lottery.

Details: 20 seats available. You write your name on a slip of paper and you can choose either 1 seat or 2 seats if you win. Almost everyone chooses 2 seats. I have never seen anyone choose 1 seat. So keep that in mind. The final count was 554 – that means 554 people wrote their names on slips of paper.

Extension:  What if you didn’t know everyone was gonna choose 2 tickets? Suppose people choose 2 tickets p% of the time. And they choose 1 ticket the other [100-p]% of the time. Write a formula for your chances of winning in terms of p.

Any other math questions you can think of?

The mathematics of music: exponential growth

Tonight I came across this lesson on TedEd:

The Physics of Playing Guitar

TL;DW: Musical notes follow exponential growth!!!!!!!

What does that mean?

Well, in the Western scale–the scale used on a piano and guitar–the frequency of each tone is (approximately 1.06) times as high as the frequency of the tone before it. Since there are 12 tones in an octave, that means that by the time you’ve jumped up a whole octave, you have multiplied by a total of twelve times.

In other words, .

Which is equal to a perfect, clean 2.  So, the frequency doubles over the course of a full octave.

For instance a standard A has a frequency of 440 Hz, which means the guitar string or piano string vibrates 440 times every second.

The next A on the piano (one octave higher) has a frequency of 880 Hz.

Double the frequency.

How amazing that the tones that share musical harmony (two A’s for example) share some sort of mathematical harmony also.

Here is a listing of note frequencies where you can check if the mathematical rule works.

Move across any row and the numbers are doubling, as we jump up by octaves.

Now move one box up, starting in any location.  This represents moving up by one tone of the Western scale.  Sure enough, the number gets multiplied by , which is about 1.06.

Understanding the Western scale opens the door to different types of scales.  There is something universal about the sound of an octave.  But why take 12 notes to get from one octave to the next?  Why not 13?

One characteristic that I have learned about Indian classical music is that the notes are less discretized, meaning a singer may move “in between” an A and a B on the scale, thereby fitting many more tones into each octave.  You can hear it in this example: the singer hits all the notes of the Western scale, but also sings around and in between them, sliding between tones continuously.