# P(Death and ICU) does not equal P(Death)

Here is a tweet I saw today.  It’s from April 7.  It’s from Tomas Pueyo, the author is this medium post on Coronavirus from March 10.  (Tomas Pueyo is not a public health expert).  When I first read this, I thought, wait that doesn’t seem right and I tried to reproduce his numbers.  To get to 36%, Pueyo is simply dividing 1.8/5 = 0.36.  But is that right?

What Pueyo is trying to do here (I think) is calculate the conditional probability that someone dies given that they are in the ICU.  That is:

$P(Death | ICU)$

By the definition of conditional probability, this is equal to:

$P(Death | ICU) = \frac{P(Death \cap ICU)}{P(ICU)}$.

So based on this calculation, Pueyo is implying that P(Death and ICU) = 0.018 and P(ICU) = 0.05.  However, he is using an estimate of P(Death) rather than the correct probability of P(Death and ICU) in his numerator.  While these numbers may be close, they are almost certainly not equal to each other:

$P(Death \cap ICU) = P(ICU|Death)P(Death)$

The only way that these two numbers would be equal to each other is if P(ICU|Death) = 1.  This seems unlikely.  There are certainly people dying from Covid-19 that never make it into the ICU.

So even if Pueyo’s number are completely correct, this calculation is almost certainly an over estimate based on basic intro probability rules.

And I’d like to add two additional points:

1. We don’t actually know what the mortality rate is of Covid-19 because we don’t have accurate counts of either the numerator (how many deaths) nor the denominator (how many cases) in a mortality rate calculation.  So this number has a ton of error in it.
2. We already know that age is a huge predictor in mortality rates for Covid-19, so to not include that in the calculation of probability of death of a specific individual of known age makes no sense.

So basically this entire calculation is meaningless.  So my advice is let’s stop playing armchair epidemiologist, especially if you can’t even get very basic probability rules correct!

Cheers.