Let me draw the calculation:
The calculation for the first ball is easy, even if you don’t have to call it Bayes to understand it, there are 1/4 of the red balls in the first pocket, and 3/4 of the red balls in the second pocket.It’s not that simple. Why write a formula?
Because sometimes you have to use a formula, especially if the total number of balls in the two pockets is different.So, I’m going to use Bayes’ formula here.
So the probability of coming from number one is 1/4, and the probability of coming from number two is 3/4.
The second ball is still red, which is equivalent to continuous occurrence, so the probability is multiplied, as shown in the figure below:
The third ball is still red, and the probability is multiplied again, as shown in the figure below:
The experiment showed that people’s direction of judgment, as well as their sense of updating their judgment based on new information, were correct.
The problem: People are underestimating the magnitude of the update.
Reaching for three red balls in a row makes that pocket 27 times more likely to be a number two pocket than a number one pocket, rather than three times as likely as our intuition.
(Note that there is still a bit of ambiguity here about multiples, but it doesn’t affect the calculation or the point of view.)
According to the above calculation, if you touch three red balls in a row, the probability of them coming from the second pocket is 27/1 +27 =96.42%.
What’s confusing about this is the probability stack.
People have a sense of one layer of probability, but when you multiply multiple layers of probability, you get a little bit confused.
In this one up here, we did three levels.
What I’m interested in is not the calculation per se, but “the speed at which it approaches the truth based on updated information”.
The predictive power of Bayesian computing.
When the brain is confronted with information, there is a process of processing.
This processing ability determines whether the brain is great or not, and whether a person can make smarter decisions in the real world.
As calculated above, for every red ball touched, the probability of a second pocket is multiplied by three.Three times, it’s 3 to the third power, which is 27 times.
Yes, this is an uncomplicated Bayesian calculation.
But you don’t see, there seems to be some kind of continuous leverage effect that allows you to draw three balls at random to achieve such high accuracy.
That’s the way awesome people think.