Lesson 2 - Mathematical thinking
These notes have been taken while studying the Crash Course: Statistics course on YouTube.
Mathematical thinking is about seeing the world in a different way. Which means sometimes seeing beyond our intuition or gut feeling.
In order to make sense of big numbers
Put them into context. For example, US dept is $20000000000000 (20 trillion). About 323000000 people live in US. So, the debt owed for each person is about $62500.
Turn a big number in to a unit of measurement. Fo example, the Kola Superdeep Borehole 40230 ft which is 7.5 miles down.
Attach a reference points for big numbers ready to go. For example, there are about 100000 words in 400 page novel. About 46000 people show up to Dodgers games in Los Angeles. A 1000000 people taking to the streets to protest might be easier to think of as 21 Dodger Stadium’s worth of people. Or 14 and a half crowds for a Real Madrid match.
- A million seconds is a little less than 12 days. 1000000000 (billion) seconds = 32 years. 1000000000000 (trillion) seconds = 32000 years.
A law of truly large numbers: The idea here is that with a large enough group, or sample, unlikely things are completely likely to happen.
The incredibly small numbers. That can be hard to comprehend.
Take the likelihood of winning a Mega Millions jackpot in the US. Right now it’s about one in 302.6 million. The probability that you’d win the jackpot is 0.000000003305.
302.6 million is the number of seconds in more than 9 and a half years.
Post by Tim Urban on Wait But Why: that’s like knowing that a hedgehog will sneeze once in the next 9 and a half years and betting on the exact second during those nine and a half years the hedgehog will need a tissue.
Though you are, by some accounts, less likely to be killed by a terrorist attack in the US committed by a refugee. In a 2016 study, researchers calculated that as a one in 3.64 billion chance in the US in a given year.
Thinking mathematically isn’t just about understanding numbers better. It’s about asking important questions about the world around us. And letting numbers illuminate those questions.
Abraham Wald. In an effort to figure out how to best protect our planes, the statisticians pour over data of the planes that returned from fighting looking at where they took damage. Where the bullet holes were. That data showed there were more bullet holes in the fuselage and fuel system and not as many in the engines. The exceptional statistician Abraham Wald studied the data and came back with the advice that surprised everyone: To put the armor where the bullet holes weren’t. Over the engines. Wald realized the bullet holes should have been evenly distributed over the planes. If fewer planes were returning with holes in the engines, that meant those planes weren’t returning home. Wald has the exceptional realization the data wasn’t a random sample of all planes. It only represented the planes that returned.
Mathematical thinking can help:
Make better decisions
See past coincidence
Help judge risks
See the broader relationships in the world