Some reflections after reading “randomness” by Deborah j. Bennett

试想一下，你站在一个领域的中间，想抓住它，你却不知道它在哪里，那么哪个方向，你会选择进行下一步？当然，任何方向你可以选择，你有相同的概率，以满足它，如果你不知道任何其他信息。 在不确定的情况下，人们如何做出决定？

Some reflections after reading “randomness” by Deborah j. Bennett

------Based on the similarity of probability and frequency
Imagine that you stand in the middle of a field and want to catch a frog which you have no idea where it is, then which direction you will choose for your next step? Of course, whichever direction you choose, you have the same probability to meet it if you don’t know any other information.  It is an indefinite event. Since it is a random event where the frog is, people seem to have no opportunity to catch the frog faster.
In an indefinite event, how do people make their decisions? First, we shall understand the similarity and difference of probability and frequency. Probability is abstract. Frequency is Specific and practical. When one event happens again and again, probability means how many times the certain situation is supposed to occur while frequency means how many times it occurs. One is that how many times it did happen, while the other is that how many times it is supposed to happen.
People usually have great interest in the probability of one event. In my perspective, it is the basic law to deal with all the random events. We can always make the denominator the number of all the situations and the numerator the number of certain situations. People are likely to screw up the denominator or the numerator.  For instance, people err in the hit-and-run problem because they see the base rate of cabs in the city as incidental rather than as a contributing or causal factor. As other experts have pointed out, people tend to ignore, or at least fail to grasp, the importance of base-rate information. In another way, it is a situation that people mistake the denominator.
People often resorted to ran-domizers for three basic reasons: to ensure fairness, to prevent dissension, and to acquire divine direction. For the three reasons, we shall emphasize more on the denominator and the numerator. Only when the denominator and the numerator are right can the result be right.
In most cases, people mistake the denominator. There are two words that A and B. If a test to distinguish A from 5 meters away whose error rate is one in a thousand has a false positive rate of 2 percent, what is the chance that a person distinguishes A actually distinguishes A, assuming there are nothing interfering?
There must be a lot of people who will answer 98 percent. The correct answer is about 5 percent. Those answering incorrectly were once again failing to take the importance of the base-rate information into account. In other words, they mistake the denominator.
The commonsense way to think mathematically about the problem is this: Only 1 time in 1000 fails to distinguish A, as compared with about 20 in 1000 who will get a false positive result (2 percent of 999). It is far more likely that any one time which tests positive will be one of the 20 false positives than the 1 true positive. In fact, the odds are 1 in 21 that anyone person who tests positive actually has the disease, and that translates into only a 5 percent chance, even in light of the positive test. In this test the true denominator is 21. And it indicates another way to state the issue is that the chances of distinguish A go from 1 in 1000 when one takes the test to 1 in 21 if a person gets a positive test result.
Sometimes people will also mistake the numerator. For instance, Mary have tossed coin for two times. We have known that one time she tossed the positive side. What is the probability that one is positive and the other is the reverse side? Answer: two-thirds.
You may be surprised that why the answer is not one-second. In fact, you will know after we list all the situations. Now we consider two sides of the coin are A and B respectively. All the situations will be AA, AB, BA, BB. If we have known A occurs at least one time. Then the denominator is three and the numerator is two. The result emerges.
As a matter of fact, people may consider a situation with randomness incorrectly because they don’t remember the basic law all the time. As long as we can consider all the situations and recognize the denominator and the numerator correctly, we can know the result. Those theorem such as Total Probability Theorem and Bayes’ Rule will bring convenience for us, but they can’t violate the basic law.


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