Your Sample May Vary

Flip a coin. Heads or tails. 50/50 chance. Heads.

Now flip it again. Heads or tails? Same 50/50 chance, right? Heads, again.

Now flip it one more time. Heads or tails. Same 50/50 chance? Are you sure?

While the possible outcomes of the coin toss remains two, and while any single toss should have that same 50/50 chance of heads or tails, does the fact that it flipped heads twice before potentially influence things. Viewed through a different lens, there's only a 1 in 8 chance you'll get heads three times in a row (2^3). This, despite the equal likelihood of one result of another on a single toss.

That coin has no memory, but the observer does.

This is the conversation I've had with many friends and family over time, and with one of my math nerd friends many times. What are the odds of something happening at any given moment? Is it the single event chance or is it the chance-over-time probability? How do we view that next coin toss? Odds of heads, or odds of three heads in a row? We can never agree.

For a more clear illustration of this, and not to condone gambling, consider the roulette strategy referred to as "doubling", for which you might be kicked off a table or out of a casino in Vegas. You may have heard it called the Martingale strategy. Minus the green 0 and 00 house advantage, a lot of people like to the think of the "roughly" 50/50 chance of getting a red or black number, and it works well here.

Bet $1 on red. Black. You lose. If you double your bet to $2 and win, you'll get your lost dollar back plus $1 in winnings. If you have to do it three times you'll need to bet $4, having lost $3. If that fails you'll be betting $8, having lost $7, etc. When you finally get red, no matter how many times you doubled, you'll only win back that first dollar. If they don't kick you off the table, at which point you've lost $1, $2, $4, $8, $16... you get the idea. I've done this to pay cab fare, never more than dollar stakes.

The trick with doubling/Martingale, is that the "risk" up front is very small, but given time, can compound dramatically. Given $1k and asked to double it or walk away broke, 62% will go broke using this strategy, most doing so when needing to cover a $512 bet. Surprisingly, 3% will be one win away from doubling before they ultimately tumble all the way down and fail.

So how does this relate to chance and probability, especially since we know that in gambling there's a house advantage? Chance will manifest per toss of a coin, or roll of dice, or spin at a table. Probability will skew results for periods, to your benefit at times and to your detriment at others, ultimately showing the same chance given enough samples. Single event, random sample, or massive dataset... it ultimately depends on the observer.

Let's talk about non-gambling chance and probability.

I'm sure anyone reading this has been stuck in traffic wondering why their lane is moving so slowly. As soon as you switch, the lane you were just in moves forward and you're stuck again. A well-known study found that, much like the "every light on the way home was red" perception, it has to do with the attention of the observer and little-to-nothing to do with reality. Lane not moving, you pay attention. Lane moving, you don't. Red light, attention. Green light, inattention. Much like chance and probability, it comes down to the observer. Of course, Mythbusters famously tested this once and, through aggressive lane changing, it was possible to move faster, but at greater risk and stress.

I've joked with my kid a lot about something I call the "Principle of Proximity". It centers around the idea that when you drive past an ATM no one is there, but when you need to use that ATM there's a line you have to wait in. Or maybe you are pulling up to an intersection and nobody is there as you approach, but by the time you arrive there's a massive 18-wheeler with a train of cars behind you have to wait for, followed by nothing. All there, together, at the same time you are; not before, and not after. It diverges into another "Confluence of Coincidence" idea, and they pair up as a cause and effect, but I digress (if you want to know more let me know).

In closing, I'll share one last personal story of chance versus probability. I have a friend who was on a plane that experienced an engine failure. The chance of that happening is slim. Shifting the point of observation from chance to probability I ask the question what are the odds the same passenger experiences TWO engine failures and I figure it's extraordinarily low. So if probability is chance over time, I ask the question: are you safer flying with that person than you would be without?