Some fun math stuff today all inspired by discussions I’ve seen online in the past week.
Another Monty Hall Explanation
I saw a discussion online recently about the Monty Hall Problem. It was a debate about the best way to explain the solution. If you need a refresher on the problem and the solutions I wrote this a while back: The Monty Hall Problem Is More Than A Game.
Many people are familiar with 99 or 1,000,000 doors explanation. I think this explanation is easier to grok after you’ve been given a more probability based explanation. The value of the 99 doors version is it highlights the beating heart of the problem — the host knows where the goats are!
Anyway, the discussion prompted me to draw out a solution I haven’t seen laid out this way before. It uses tree thinking (big surprise right?).
X² and its neighbors
What’s bigger:
20 x 20 or 19 x 21?
And is your answer generally true.
This is the same as asking, what’s bigger:
X² or (x-1)(x+1)?
Somewhere in your mental library, there’s a dusty text probably wedged between the book called “Polynomial” and the volume named “Completing the Square” that will remind you that (x-1)(x+1) = X² – 1
Which means your answer generalizes:
400 > 399
X² > X² – 1
So just knowing what 20² is means you immediately know what 18 x 22 is 396, X²-4.
The sandwiches around “20” continue down the squares:
17x 23 = 400 – 9
16 x 24 = 400 – 16
15 x 25 = 400 – 25
Et cetera
Clever Polling
This is neat:
I posted that I remembered reading about a sampling technique a long time ago for how to poll for things that people wouldn’t answer truthfully (ie “Have you cheated on a test/spouse/taxes?”).
@LoneSands recognized what I was referring to:
The version I saw had you flip a coin twice before answering an agree/disagree question. If first flip was heads then answer honestly; on tails use the 2nd coin flip to decide.
Ah yes, that was it. So how do you use this tree math to infer the “true” response rate assuming people follow the rules?
This is how I solved it….I just used some real numbers.
Suppose n = 1000 and I had 550 agree.
I expect that 250 are “agree” and 250 are “disagree” because of the second coin flip.
That means 300 people actually agreed.
The true agree rate then is 300/(200 disagrees + 300 agrees) = 60%
LoneSands gave the general formula:
True agree rate = 2*(observed agree rate) – .5
That’s friggin’ tight. I like that.
I wanted to bridge the general solution to mine for validation. This is a great opportunity to demonstrate how to use GPT for self-tutoring.
I gave GPT the general formula and then prompted:
I needed to push a bit more because the bridge wasn’t complete. Luckily, GPT never gets tired of my tediousness.
I’ll tell you a secret.
I’m taking Algebra 2 on Khan Academy right now. These notes remind me what has been refreshed or reinforced (I’m about 2/3 finished with the course):
My plan is to re-do all my HS math through Calc BC and then assess what math I want to learn for the first time after that. It’s really different to take math as an adult because:
a) There’s no deadlines so the stress is gone
b) I actually care beyond a test score
c) Re-learning math after you’ve spent 20 years doing math at work (usually through heuristics and shortcuts passed down), you naturally try to draw connections between the math class examples and the problems in your field. For example, deriving the geometric series in Algebra 2 is synergistic with thinking about compounding problems. But the first time I took Algebra 2 I surely could care less about how much total distance a monkey swinging from a branch with a starting arc length of 50 ft who loses 15% per swing covers after 5 swings.
Math scratches the same itch as NY Times puzzle games for me, which is to say I don’t need an instrumental reason to do it. But it’s also a convenient discovery to hear one of the the late Charlie Munger’s big pieces of advice (amongst many amazing bits of advice) to be “pay attention in HS math”.
☮️
Stay Groovy