can you trust your judgment with quick math calculations?
Can you trust your judgment with quick math calculations?
Suppose you undergo a medical test for a relatively rare cancer. The cancer has an incidence of 1 percent among the general population. Extensive trials have shown that the reliability of the test is 79 percent. More precisely, although the test does not fail to detect the cancer when it is present, it gives a positive result in 21 percent of the cases where no cancer is present - what is known as a false positive. When you are tested, the test produces a positive diagnosis. What is the probability that you have the cancer?
[The Unfinished Game by Keith Devlin page 140]
Spoiler:
If you are like most people, you will assume that if the test has a reliability rate of nearly 80 percent, and since you have tested positive, the likelihood that you do indeed have the cancier is about 80 percent (i.e., the probability is approximately 0.8). Are you right?
No. Given the scenario I just described, the likelihood that you have the cancer is just 4.6 percent (i.e., the probability is 0.046). There is a less-than-5-percent chance that you have the cancer.
...
Here is how you arrive at that figure:
P(H) = 0.01
P(E|H) = 1 (the test always shows positive if the cancer is present)
P(Hwrong) = 0.99 (99 percent of the population is cancer-free)
P(E|Hwrong) = 0.21 (the test gives a false positive in 21 percent of cases)
So by Bayes' formula
P(H|E) = 0.01*1 / ((0.01*1) + (0.99*0.21))
= 0.01 / 0.2179
= 0.0459
[The Unfinished Game by Keith Devlin pages 140 to 141]
Probability and statistics should be core material, no matter what. It can come after algebra, or concurrent. It's that important. We teach exponential growth and the interest rate equation: P*exp(r*t). Introductory prob/stats is no more difficult to understand and equally important.
I'm fine with a lamed-down version for high school and another less lamed down version for non-math majors at college level.
Understanding the meaning of a statistic is supremely important when they are thrown at us by our politicians on a constant basis.
Seriously. I was on the advanced math track in middle school and I was in algebra is 8th grade. Fresh outta honors geometry. I didn't get into proper statistics until I was in college and I didn't understand it until some years later. It is the door keeper between what we can measure and what we can know. I'd love for middle-schoolers to get it, but no one really gets it. That's the difficulty.
My #1 problem with the standard math curriculum right now is that it's geared from day one towards getting students to calculus when it should really be geared from day one towards getting them to statistics.
And before some beta bitchboy aspie case starts arguing with me, we're talking about all students, not the two percent or whatever that will actually use calculus for some real purpose in their fucking lifetimes.
It doesn't have to be difficult. First teach the terms. Have them draw rough sketches of bell curves. Have them take some data from somewhere and show that even when they all take the same data, they get a range of results. Show how that range is like a bell curve. Talk about variance and confidence intervals. Play some dice game and talk about the probable outcomes of 2d6. Have them roll more and more dice and show how the data of the sum of the dice rolls approaches a bell curve.
Nothing about learning has to be difficult.
Prob/stats does involve calculus, but so does calculating the slope of a line. I bet you knew how to tell the slope of a straight line before you knew any real calculus, though.
I bet you know about that trick where you can make a curved line by drawing a bunch of straight lines just so:
Did you know that is called a Bezier Curve of 2nd order?
Did you know that the curvy line tool in MS Paint draws a Bezier Curve of 3rd order?
Maybe so. Plenty of people don't know these things, but they are familiar with both.
My point is that you don't have to get into the nitty gritty numbers and stuff to get a working understanding of the concepts at play.
I definitely agree. Geometry is taught because it's ancient and useful in forming right angles. Calculus is taught because Newton and Leibniz were motherfuckers who understood motion. Trig and complex analysis are important for understanding all the bullshit that goes into that computer you're typing on, but statistics and probability are how you bridge belief with reality and that's very difficult. You gotta learn so much before you can understand what you're doing.
The point is that you don't get statistics. You can understand it. You can show the proof of how if you have 25 people in a room, chances are very good that someone shares a birthday. You can even show that your dataset hits that heralded p-value of .05. But unless you're gonna pencil and paper every single opportunity for stats and probs in your life, you'll see that your brain isn't meant for this. You don't live life like Zeckhauser plays backgammon.
There's a reason why fivethirtyeight.com is so in demand these days.
Common sense and 5 seconds of mental arithmetic tells you it's close to 1/22.
UK schools aren't big on stats at all, certainly when compared to algebra and trig. That's probably why the majority jump to the wrong conclusion whenever they see a "fact" published by the gutter press over here.
Common sense and 5 seconds of mental arithmetic tells you it's close to 1/22.
UK schools aren't big on stats at all, certainly when compared to algebra and trig. That's probably why the majority jump to the wrong conclusion whenever they see a "fact" published by the gutter press over here.
If any arguments depends on "common sense" you are in trouble. I believe Voltaire said that common sense ain't so common.
Probability and statistics should be core material, no matter what. It can come after algebra, or concurrent. It's that important. We teach exponential growth and the interest rate equation: P*exp(r*t). Introductory prob/stats is no more difficult to understand and equally important.
Ironically, over here statistics is considered the "easy" maths, and algebra the "hard", so since I was good at maths, I always did algebra and not statistics, so I have a sub-major in maths from uni and don't know squat about probability and statistics.
can you trust your judgment with quick math calculations?
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Originally Posted by Renton
