Probability is a matter of perspective

One of my favorite hobbies is predicting outcomes, and my dirty little secret is that I like to be as ignorant as possible when I’m doing it (assuming it’s not influencing an important decision). When I rely more on deduction than on facts, I feel like I’m beating the odds.

This is only partially false.

“Probability” is a function of your knowledge and perspective

Nothing in probability can be intuitive before you grasp this fundamental idea, for which I have a handy little demonstration.

Imagine I flip two quarters and catch one in my closed right hand, and the other in my closed left hand. Then I pose to you a question: What are the chances that both quarters came up heads?

Despite American public education, you can probably figure out the answer is one out of four. That is, of the four possible outcomes

1. Right hand: heads, Left hand: heads
2. Right hand: tails, Left hand: heads
3. Right hand: heads, Left hand: tails
4. Right hand: tails, Left hand: tails

only one has both quarters coming up heads.

But imagine I also told you that at least one was heads for certain. Using this information, what are the chances that both are heads? Most people will guess one in two, but I’ve actually only ruled out the fourth possibility:

1. Right hand: heads, Left hand: heads
2. Right hand: tails, Left hand: heads
3. Right hand: heads, Left hand: tails
4. Right hand: tails, Left hand: tails

So the chances are one in three.

But say I phrase the new information a little differently, by saying that the quarter in my right hand is heads. This is different than what I said before, because it eliminates two possible outcomes.

1. Right hand: heads, Left hand: heads
2. Right hand: tails, Left hand: heads
3. Right hand: heads, Left hand: tails
4. Right hand: tails, Left hand: tails

So the chances are one in two now.

This illustrates the counterintuitive part: the type of information you received changed the probability. It matters that I specify which hand certainly has a coin that is heads. That means that these statements

“There is at least one coin that is heads.”
“There is at least one coin that is heads in my right hand.”

are not mathematically equivalent! This is just the sort of philosophical insight that makes probability very interesting to me. There are also a lot of philosophical consequences to this idea. I bet I could spend my entire life writing about how probability is related to the idea of limited knowledge, and what this tells us about God and the universe we inhabit.

Self-congratulatory fap fap

I’m pretty good at building systems in this way, preferring insight and deduction to data. I rely on very few principles and a few aesthetic ideas, and the rest follows logically. I make more amateur (stupid) mistakes that way, but minimalism and abstraction also give me a better grasp of the fundamentals. And often, a unique perspective has intangible benefits. As you can see.

About Aeoli Pera

Maybe do this later?
This entry was posted in blather, economics, life tips, math. Bookmark the permalink.

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