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re: Memorylessness at the Bus Stop: Using R on transit data to test a hypothesis VIEW POST

FULL DISCUSSION
 

I challenge you to a game. You win when you flip a "fair" coin and get Heads.

You flip the coin...and get Tails.

But there's a loophole in the game rules. You win when you flip Heads, but you can flip that coin as many times as you want until you get Heads. So you flip the coin, over and over, until you finally get Heads and win the game. As long as the game isn't rigged, there's always the possibility that you might get Heads on the next flip.

But flipping a coin is a rather tedious process. You have to get a coin, move it up in the air, bring it back into the ground, and check whether it is Heads or Tails. What if you could flip multiple coins at the same time, and then check them all at once? There's no real penalty for flipping more coins than necessary, after all.

The odds of flipping one coin and getting at least one Heads is 50%.

The odds of flipping two coins and getting at least one Heads is 75%.

The odds of flipping three coins and getting at least one Heads is 87.5%.

The odds of flipping ten coins and getting at least one Heads is 99%.

Etc.

Improbable events can still occur (such as ten coins flipping tails), and you have to be ready for them, but the Law of Large Numbers suggest that the odds are in your favor...even if each individual flip is memoryless.

Waiting for the bus stop is like playing my game. Every minute that you wait, you're playing the "game", hoping that the memoryless bus will come and you "win". When you wait another minute, you're still playing. And if you decide ahead of time that you're willing to wait 20 minutes for a bus stop, you're merely precommitting yourself to playing the game for a certain period of time. Obviously you can win at any time, but you're not betting on winning on the first round. You're betting that you'll win before the 21st round. You're willing to keep playing until you win (or your "waiting" budget runs out and it's better to take a taxi to arrive at your destination since that'd probably be quicker than to keep playing the game).

It is not futile to wait. You just have to know the rules.

 

I agree with everything you've said -- the game you describe is an instance of the geometric distribution, which is memoryless and the discrete case analogue to the exponential distribution. And yes, committing yourself to waiting for 20 minutes as opposed to 5 or 10 will by definition expose you to a higher probability of the bus coming.

The point of my article, and the thing about memoryless that for me at least is fascinating and spooky, is that your odds never improve even after waiting. It's a counterintuitive idea -- you'd think that since a bus must be on the way, your chances should be improving with every passing minute, when in fact they are not. Personally, I find that pretty cool!

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