The Montihall problem, which once puzzled many mathematicians, seems to have pushed probability and intuition against each other. In this case, we are faced with choices that seem to violate the basic principles of probability, but at the same time it feels as if there is some kind of "attractor" guiding our decision-making.
First, let's review this classic question. Let's say we have 3 doors and there is a prize behind one of them. You choose one of the doors, but the host opens the other two doors with no prizes. At this point, should you choose the remaining door?
This question seems to go against the basic principles of probability. From a probability point of view, the probability that each door will be chosen is 1/3. However, when the host opened the other two doors without prizes, the probability of the remaining door seemed to increase. Many even believe that the remaining door has a 2/3 chance of containing a prize.
However, this is actually a misleading. When the host opens two doors without prizes, the probability of the remaining door is still 1/3. This is because the probabilities are independent of previous events. Even if we know the previous result, it doesn't affect the probability of the remaining door.
So, why do we have this feeling of "attractor"? It may be because we often encounter similar situations in our lives. When we make decisions, we are often influenced by previous results. For example, if we toss a coin 10 times in a row and all heads are heads, we may assume that the probability of tails on the next coin toss is higher.
However, this thinking is wrong. Each coin toss is independent, and the probability of heads or tails on the next coin toss is 1/2, regardless of the previous outcome. Similarly, in the Montihall problem, the probability that each door will be selected is 1 in 3, regardless of the previous outcome.
So, how should we deal with this feeling of "attractor"? First, we need to recognize that this intuition may be wrong. Second, we need to analyze the problem rationally and not rely on intuition alone. Finally, we need to keep an open mind and embrace new perspectives and ideas.
When confronted with the Montihall problem, we must not only understand the conflict between probability and intuition, but also learn how to apply this knowledge in real life.
First of all, we need to understand that probability is not an absolute value, but a relative concept. In the Montihall problem, each door has a 1/3 chance of being chosen, but this does not mean that the remaining door has a 2/3 chance of containing a prize. Therefore, we need to analyze the problem rationally and not be swayed by intuition.
Second, we need to recognize that probability is independent of previous events. Regardless of the previous result, it will not affect the probability of the remaining door. Therefore, even if we have been positive 100 times in a row in the previous trial, it does not assume that the probability of a negative next time is higher. Because each coin toss is independent, the probability of heads or tails in the next coin toss is 1/2.
Finally, we need to recognize that probability is a long-term trend, not a short-term outcome. In the Montihall problem, if we do 100 experiments, we might find that 99 times it is all heads and 1 is tails. But that doesn't mean that the remaining door has a 2/3 chance of containing a prize. Because this is only a short-term result and does not represent a long-term trend.
In real life, we also need to apply this knowledge to make more informed and accurate decisions. For example, in the field of investment, we need to analyze market trends and risks rationally, and not be swayed by short-term fluctuations. At the same time, we also need to recognize that the market is independent, and do not assume that future market movements will be affected because of previous market movements.
In conclusion, the Montihall problem reveals the conflict between probability and intuition. By analyzing problems rationally and keeping an open mind, we can better understand and respond to such conflicts. Let's demystify the Montihall problem and use scientific methods to guide our decision-making!