In exploring randomness and probability theory, one thought-provoking phenomenon is the interaction between the interchangeable sides of the fair dice and human cognition. When the six sides of the dice are exactly equal and indistinguishable, we tend to assume that all outcomes are equally likely. In practice, however, in order to clearly distinguish between different outcomes, we must mark these faces – as in the case of the dots on standard dice, the sum of the dots on opposite faces is 7.
Historically, this notion led the German mathematician and philosopher Gottfried Wilhelm Leibniz to make a famous mistake. In 1714, in a letter he explained to the scholar Louis Bourg the basic concepts of classical probability, but ignored the orderliness of the combinations. He mistakenly thinks that rolling two dice to get 12 is the same as getting 11, when in reality there is only one combination of 12 (6,6) and 11 has two combinations (5,6 and 6,5).
This misconception stems from Leibniz's view of indistinguishable alternatives as equivalent. For example, if two identical dice D1 and D2 are not distinguished, one might mistakenly think that rolling (5,D1) and (6,D2) and (5,D2) and (6,D1) are the same. But as long as you give the dice a different color, such as D1 in red and D2 in blue, this confusion will be eliminated immediately.
Coincidentally, one of Leibniz's own philosophical principles is the "principle of discernible identity", which makes this delusion caused by indiscernibility coincidentally called the "Leibniz illusion".
So, returning to the modern probability puzzle Montihall problem, are we also falling into a cognitive trap similar to Leibniz 300 years ago? Perhaps, just like labeling the dice with different colors, by clearly labeling the three doors, we can avoid falling into this mistake of being based on intuition rather than strict logic, and thus understand and solve such complex probability problems more accurately. When faced with random phenomena, the key to solving such puzzles is to keep a clear awareness and distinguish between events that seem to be the same but actually have different probabilities.
In the Montihall problem, we are also confronted with alternative possibilities that appear to be equivalent but are actually of different probabilities. After initially choosing a door, the host reveals a door that hides a goat, at which point the remaining two doors do not have the same probability of owning a car. Intuitively, though, there seem to be two and only two possible outcomes: the door you initially chose or the door that the host didn't open. However, this superficial symmetry obscures the true nature of the probability distribution.
When the host opens a door, he offers new information – namely that he has ruled out a false answer. This means that the probability that your initial choice is correct has not changed (it is still 1/3), and the probability of a car hiding behind the remaining unrevealed door has increased to 2/3. The key here is to distinguish and understand how each possible event occurs and how its probability is updated.
Just like color-labeling dice, if we can clearly distinguish and track changes in the state of each door, we can avoid falling into the trap of the "Leibniz illusion". In fact, in the Montihall problem, we need to adjust our beliefs based on known conditions, rather than simply insisting on a consistent intuitive judgment.
This example vividly illustrates the importance of accurately distinguishing, considering all possibilities, and updating information in a timely manner when understanding and applying randomness and probability theory. By delving into such cases, we can better identify and overcome cognitive biases and improve our ability to make rational judgments in complex real-world decisions.
In summary, both the misjudgments of the ancient mathematician Leibniz and the probability puzzles of modern entertainment point to a central idea: when dealing with random events, we cannot judge possibilities based on our feelings alone, but need to rigorously analyze the situation and ensure that we correctly identify and quantify all alternative possibilities. In this way, we can truly dispel the cognitive misunderstandings caused by confusion and misunderstanding randomness, so as to more accurately grasp the operation law of the uncertain world.