My Last Vaccination Post For A While

It’s preventing me from getting other things done.  However, I posted a comment over at Phil’s blog that has as much to do with numbers as the topic of vaccinations, and it illustrates an interesting version of the Gambler’s Fallacy, so I’m replicating it here:

A comment read, in part:

That goes for every parent that has witnessed the change in their child immediately after a vaccine or shortly thereafter.

My response:

This actually *is* a fallacy, a post hoc ergo propter hoc. There is a simple reason why this is not relevant, take the following facts…

* children take vaccines
* autism displays its first symptoms in childhood
* children under the age of 5 make up ~7% of the population
* there are ~360 million people in the U.S.
* about 80% of children are vaccinated entirely

(editor’s note: I didn’t make those numbers up, you can find them with a couple seconds and a web browser)

This means 360 x 0.07 x .8 = 2 million children (roughly) have been vaccinated. With the vaccination schedule being what it is, then, there are somewhere around 100,000 children getting a shot every month (that last one is handwavy, it assumes a lot about frequency distributions, but that’s not really germane to my point). Autism rates are estimated at anywhere between 1 in 100 and 1 in 150 children, that means we have about 17,000 diagnosis of autism. If every single one of those autism diagnosis was given to a vaccinated child (they’re not, but again for our sake here it introduces very small error), and those 17,000 have a scatter distribution of vaccination patterns, that means not one, not dozens, not hundreds, but *thousands* of those diagnosis came within days or weeks of a vaccination.

Put those thousands of people together on a message board (and since autism is hard to deal with, a very high percentage of these family *do* bond together, like SMA sufferers or MS or cancer or any other family-impacting disease), you’ll have a few thousand people all saying to each other, “Gee… MY kid got a shot right before her symptoms started showing, too! There are thousands of us! THAT CAN’T BE A COINCIDENCE.”

But you can see, it actually *isn’t* a coincidence… it’s exactly what we would expect to happen.

This is one of those times where people don’t think about what big numbers really mean.  Here’s another example, one without a muddying emotional component: the odds of hitting “black” 20 times in a row on a standard “0” and “00” roulette wheel are really bad.  A roulette wheel has 18 blacks and 18 reds numbered 1-36, plus green 0 and 00 for a total of 38. The probability of a black is 18/38,  the probability of black 20 times in a row is

(18/38)^20 =~ 3,091,874 to one.

Holy bejeezus, you think, that’s crazy impossible!  For the record, it’s still much better than the odds of winning the California lottery.

Here’s the thing.  If every man, woman, and child in the United States started playing roulette right now, guess what?  After 20 spins each,  we’d have roughly100 people (give or take) staring thunderstruck at the roulette wheel, amazed at their unbelievable luck.  Put all of those 100 people on the same message board, and they’d probably attach some sort of crazy significance to the day that they all won – it must be significant, how could *that* be a coincidence?   Not to burst their collective bubble, but there’s nothing amazing or unbelievable about it… it’s what we expect to happen.

[edited to add] apologies for the above, I made a reference error (as was pointed out here and here).  Currently, according to the Census Bureau, the number of people in the U.S. is ~306 million, not ~360 (http://www.census.gov/main/www/popclock.html).  Just goes to show that transpose errors happen to the “best” of us, right?  I’ll leave correcting the results above as an exercise for the reader; the end result is not significantly different.