Sharpening Intuition with Cards? Here Comes the Math...

So, I've been working on my sixth sense quite a bit recently. One of the things I've been doing is the classic exercise of guessing red or black with a deck of cards.

On the face of it, it's pretty easy. 26 red and 26 black, so you should hit about 50%, right?

So, I practiced. I had varying levels of success, including a couple spikes into the 40s, but usually around the 29-32 range. 29-32 right out of 52 is more than expected, but is it beyond the level of statistical "noise"? Putting on my math geek hat, I decided to investigate.

I won't bore you with the pen-and-paper route I used; we'll use one of my new favorite toys: Wolfram Alpha.

Ok, so, each card is an event in a series of 52. The probability of getting any given card right is 50%. The tool to use for this is a "binomial distribution", and you can take a look here:

http://www50.wolframalpha.com/input/?i=binomial+distribution+n%3D52+p%3D.5

The important number to look at in the results is the "standard deviation". Simply put, everything inside the standard deviation accounts for 68.2 percent of cases. Here, the standard deviation is (for our purposes) 3.6. This means that on average, I'll score a little outside the standard deviation of a single run, as 26+3.6=29.6.

I started to sweat a little bit on this, as I seemed pretty close to the standard deviation. However, I remembered my range is 29-32. It is rare for me to score lower.

This has two related factors: First, I'm consistently scoring on the good side of the bell curve, with dips on the downside very rare (happens once in a great while).

Second, and relatedly, we can look at this not as simply a series of 52 events, but as aggregate data. Sorry, I'm geeking out too much; lemme 'splain:

Say you scan a deck of cards twice over the course of a day. Say you get 31 (beyond standard deviation) for the first run, and let's say 29 (just under standard deviation) for the second run. The thing is with statistical data, the greater the number, the sharper the picture. Let's crank up the definition of the picture and combine them.

When you increase the sample size, the standard deviation doesn't increase at the same rate: for 104 cards, the standard deviation isn't 7.2... it's 5! Don't believe me? Check it out:
http://www50.wolframalpha.com/input/?i=binomial+distribution+n%3D104+p%3D.5

So let's say you do the same the next day. Now, what we have here isn't a case where we're a little beyond the standard deviation and a little under, we're waay over: 60 (number of cards right - 5 (standard deviation) - 52 (the median number of cards) = 3 beyond the standard deviation. Not only are you into the realm of statistical significance, you're 3/5ths of the way beyond it!

I know, I know, it may seem silly to get into the itty bitty numbers to dredge up a trace of data. But here's the thing: Numbers don't lie. I may not be far into the realm of statistical significance, but by the same token, I'm definitely showing that there's something going on.

Want to stretch your brain a little more? Suppose you keep going, and double it again. Now we're going from a portable black-and-white TV to a High-definition LCD:
http://www50.wolframalpha.com/input/?i=binomial+distribution+n%3D208+p%3D.5

Assuming that you're doing about the same all the way through, we're looking at 120 right out of 208. The median (middle point) is 104, with a standard deviation of 7.2. 120 cards right - 104 median = 16. That's two standard deviations away from the median. That's getting into the tail end of the bell curve, and statistically significant by anyone's standards.

Now, I have to acknowledge the nay-sayers and cynics who will (rightly) say that the last two thirds of my examples are projections, and that this is subject to cherry-picking, as I didn't include my dips in this example. They have a small point, but they're also missing the big picture, and that is this: If you're using this as a means of training your intuition/6th sense, pay attention to the small stuff. It adds up faster than you might think.