Why probability isn’t real
There’s a probability paradox called the “Sleeping Beauty Paradox”. It seems simple at first, but philosophers and statisticians still haven’t figured it out.
Here’s how it goes. An experimenter flips a coin on Sunday. Then, he puts his experiment subject, Sleeping Beauty, to sleep. When Sleeping Beauty wakes up, she won’t remember if the coin was heads or tails, and she won’t remember if she’s been woken up before.
If the coin’s heads, then the scientist wakes up Sleeping Beauty on Monday.
If the coin’s tails, then the scientist wakes up Sleeping Beauty on Monday, puts her back to sleep, and then wakes her up again on Tuesday.
So, suppose you’re Sleeping Beauty, and you wake up. You have no idea how the coin flip went and you have no idea what day it is. What’s the probability that the coin came up heads?
The paradox
There are two ways of looking at this problem. They both make sense when you explain them. They both seem equally valid. And they both give you completely different answers.
One way of looking at the problem is looking at the coin flip. The coin had a 50/50 chance of being heads. And you’re gonna wake up either way — the fact that you’re awake now gives you no new information about the coin flip. So the coin is still 50% likely to have come up heads.
Another way of looking at the problem is by looking at the fact you’ve woken up. Suppose you ran this experiment 100 times. Sleeping Beauty would wake up 100 times in a world where the coin was heads, and 200 times in a world where the coin was tails. So if you’re Sleeping Beauty and you’re waking up, there’s only a 1/3 probability that the coin was heads.
Obviously, the probability can’t be 1/2 and 1/3. It’s a paradox. So what the hell’s going on?
Why Blackjack Is Easier When You’re Evil
In the movie Austin Powers, Austin Powers sits down at a blackjack table next to Dr. Evil’s evil henchman, Number 2.
The dealer deals Number 2 a king and a 7, making 17.
Generally when you have 17 in blackjack, you shouldn’t ask for any more cards. But Number 2 is a special case. When he does it, it’s a smart move.
That’s because Number 2 has a special X-ray eye patch. He sees that the top card of the dealer’s shoe is the 4 of clubs. So he says “hit me” — and makes 21.
What happened? The 4 was always on top of the deck. If Austin Powers had been in Number 2’s chair, gotten 17, and asked for another card, he would’ve made 21.
When Number 2 has 17 and says “hit me”, it’s a smart move. But if Austin Powers has 17 and says “hit me”, it’s a dumb move. Why?
Because Number 2 has more information than Austin Powers. Number 2 knows exactly what’s gonna happen when he says “hit me”. He doesn’t have to guess what card is on top of the deck. He already knows.
Whereas Austin Powers is just guessing. He doesn’t know there’s a 4 on top of the deck, so he has to use probability.
And probability is great when you have limited information. But it’s way better to just know what the top card of the deck is.
Randomness
When we say “random”, what are we talking about?
In theory, something is “random” if nothing causes it. What causes a coin to be heads or tails? We say it’s “random”
But nobody’s ever discovered anything that was truly “random”. Most of the things we call “random” are actually chaotic.
Suppose you roll a die. We call that “random”. But it’s not actually random. The number that comes up when you roll a dice is caused by how much velocity and angular momentum you roll it with, plus the initial position of the die
If you roll a 6, it’s because you rolled the die with a velocity and angular momentum that combined to make a 6 show up. If you threw the die slightly faster or slightly slower, or put a little more or less spin on it, it wouldn’t have been a 6.
However, a dice roll is random for all human intents and purposes. That’s because it’s pretty tricky to deliberately roll a 6. You can’t really control the outcome of a dice roll. So when we need to make something random, a dice roll is good enough.
Same thing with a deck of cards. The card on top of the deck isn’t “random”. It’s there because it got put there by the outcome of the shuffle. But when you shuffle a deck of cards, it’s pretty hard to predict the order that the cards will show up in. So we call a shuffled deck of cards “random”.
So What The Hell Is Probability?
Did you know that there’s no such thing as color?
Color represents light wavelength. When we look at a firetruck, a stop sign, or a glass of pinot noir, we see the color red. But all that means is that firetrucks, stop signs, and pinot noir emit a light wavelength around 700 nanometers.
If you looked at them with a radio telescope, you wouldn’t see them as “red” anymore. Without humans to see them, they wouldn’t be “red”. They only become red when your brain processes the image from your eye.
Probability is the exact same thing. If I flip a whole bunch of coins, half of them will be heads. So we say the probability of flipping heads is 50%.
But in the real world, when you flip a coin, it’s not gonna come up “50% heads” — it’s gonna come up either heads or tails, based on the force and spin you flip it with. If you flip it with the same force, spin, and initial position every time, then it will always be heads, or it will always be tails.
Coin flips are too chaotic for us to measure. So we do the next best thing, and we guess. And the best way to guess is to use probability.
That’s why probability changes based on how much information you have. For Austin Powers, the probability of hitting a 4 to make 21 is 1 in 13. But for Number 2, playing with the exact same hand and sitting in the exact same chair, the probability of hitting a 4 to make 21 is 100%. The actual real life situation didn’t change — the only difference is that Number 2 knows more than Austin Powers does.
This is how you end up with the Sleeping Beauty paradox, where the probability of a coin being heads can be both 0.5 and 0.33 at the same time.
The coin will be heads half the time, so the probability of the coin being heads is 0.5. However, 2/3rds of the time Sleeping Beauty wakes up, the coin is tails, so the probability of the coin being heads is also 0.33. The real life situation is the same. But there are 2 different reference frames you can use to look at the problem, and they result in 2 different probabilities.
Same thing with the Austin Powers problem. For Austin Powers, the probability of hitting a 4 is 1 in 13. For Number 2, the probability of hitting a 4 is 100%. The real life situation is the same. But there are 2 different reference frames you can use to look at the problem, and they result in 2 different probabilities.
Theoretically, if you knew everything there was to know, you wouldn’t need to use probability. If you have all the information in the universe then you can predict everything with 100% accuracy.
For example, you can calculate how much force and angular velocity someone will use to roll a die, and you can figure out what the result of the die roll will be in advance.
But for us mere mortals, probability is still useful. Color isn’t real, but being colorblind still handicaps people. Similarly, probability isn’t real, but understanding probability helps you make better decisions in the face of uncertainty.
Hi! Thanks for reading.
If you’re new here, my name’s Theo. I write articles about subjects that interest me, like social sciences, the books I read, and the best ways to learn stuff.
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