Thread by @PenLlawen

I want to talk about this for a second, and how it's nonsense.

The manufacturer, BD, claims a 16% false positive rate for these tests [1] (the false negative rate is, they claim, approximately zero.)

[1] fiercebiotech.com/medtech/bd-net…

1/

https://twitter.com/elonmusk/status/1327125840040169472

So if you have COVID-19, and you take one test, you have a 16% chance of a negative result (and a 84% of a positive one.)

If you take two tests, the chances of both tests coming back positive is (0.84 * 0.84) = 70.6%. You have almost a 1/3rd chance that one result is wrong.

2/

If you take four tests, the chances of all four tests coming back positive is (0.84^4) = 49.8%. So you have a coin-flip 50/50 chance, more or less, of one or more of your tests coming back negative, even though you have the disease.

3/

What about Musk's scenario? Well, the maths is a little more complicated, but it goes like this. First, we figure out the chances of a specific combination of outcomes, for example two positive tests, then two negative tests.

That's (0.84✅*0.84✅*0.16❎*0.16❎) = 1.81%

4/

But that's only one way that you could get two positive and two negative results from four tests. We need to consider all the other ways that could happen. The maths here is called combinations; we can calculate there are 6 ways to choose 2 elements from a set of 4.

5/

I'll write them out longhand to show you the options. A tick is a positive test, a cross is a negative test.

✅✅❎❎
❎✅✅❎
❎❎✅✅
✅❎✅❎
❎✅❎✅
✅❎❎✅

6/

So there are six ways that someone with COVID-19 could get two positive and two negative results. Hence, on a test with a 16% false positive rate, the chance of that occurring is (1.81*6) = 10.9%. Quite high.

7/

So to summarise: a thing happened that has an 11% chance of happening, and Musk, this much-fêted intellectual titan, concluded that this was "extremely bogus" and tweeted it out to rapturous applause from COVID deniers.

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PS I have "false positive" and "false negative" inverted in the first tweet in this thread (but the correct way around elsewhere.)

thx @RDRv3

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