時間序列︰賭徒謬誤

賭徒謬誤是什麼?

Gambler’s fallacy

The gambler’s fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the mistaken belief that, if something happens more frequently than normal during some period, it will happen less frequently in the future, or that, if something happens less frequently than normal during some period, it will happen more frequently in the future (presumably as a means of balancing nature). In situations where what is being observed is truly random (i.e., independent trials of a random process), this belief, though appealing to the human mind, is false. This fallacy can arise in many practical situations although it is most strongly associated with gambling where such mistakes are common among players.

The use of the term Monte Carlo fallacy originates from the most famous example of this phenomenon, which occurred in a Monte Carlo Casino in 1913.[1][2]

Simulation of coin tosses: Each frame, a coin is flipped which is red on one side and blue on the other. The result of each flip is added as a colored dot in the corresponding column. As the pie chart shows, the proportion of red versus blue approaches 50-50 (the law of large numbers). But the difference between red and blue does not systematically decrease to zero.

 

為何會深值人心裡??

Psychology behind the fallacy

Origins

Gambler’s fallacy arises out of a belief in a “law of small numbers“, or the erroneous belief that small samples must be representative of the larger population. According to the fallacy, “streaks” must eventually even out in order to be representative.[7] Amos Tversky and Daniel Kahneman first proposed that the gambler’s fallacy is a cognitive bias produced by a psychological heuristic called the representativeness heuristic, which states that people evaluate the probability of a certain event by assessing how similar it is to events they have experienced before, and how similar the events surrounding those two processes are.[8][9] According to this view, “after observing a long run of red on the roulette wheel, for example, most people erroneously believe that black will result in a more representative sequence than the occurrence of an additional red”,[8] so people expect that a short run of random outcomes should share properties of a longer run, specifically in that deviations from average should balance out. When people are asked to make up a random-looking sequence of coin tosses, they tend to make sequences where the proportion of heads to tails stays closer to 0.5 in any short segment than would be predicted by chance (insensitivity to sample size);[10] Kahneman and Tversky interpret this to mean that people believe short sequences of random events should be representative of longer ones.[9] The representativeness heuristic is also cited behind the related phenomenon of the clustering illusion, according to which people see streaks of random events as being non-random when such streaks are actually much more likely to occur in small samples than people expect.[11]

The gambler’s fallacy can also be attributed to the mistaken belief that gambling (or even chance itself) is a fair process that can correct itself in the event of streaks, otherwise known as the just-world hypothesis.[12] Other researchers believe that individuals with an internal locus of control—i.e., people who believe that the gambling outcomes are the result of their own skill—are more susceptible to the gambler’s fallacy because they reject the idea that chance could overcome skill or talent.[13]

 

幾乎無法抹除耶!!

Possible solutions

The gambler’s fallacy is a deep-seated cognitive bias and therefore very difficult to eliminate. For the most part, educating individuals about the nature of randomness has not proven effective in reducing or eliminating any manifestation of the gambler’s fallacy. Participants in an early study by Beach and Swensson (1967) were shown a shuffled deck of index cards with shapes on them, and were told to guess which shape would come next in a sequence. The experimental group of participants was informed about the nature and existence of the gambler’s fallacy, and were explicitly instructed not to rely on “run dependency” to make their guesses. The control group was not given this information. Even so, the response styles of the two groups were similar, indicating that the experimental group still based their choices on the length of the run sequence. Clearly, instructing individuals about randomness is not sufficient in lessening the gambler’s fallacy.[19]

It does appear, however, that an individual’s susceptibility to the gambler’s fallacy decreases with age. Fischbein and Schnarch (1997) administered a questionnaire to five groups: students in grades 5, 7, 9, 11, and college students specializing in teaching mathematics. None of the participants had received any prior education regarding probability. The question was, “Ronni flipped a coin three times and in all cases heads came up. Ronni intends to flip the coin again. What is the chance of getting heads the fourth time?” The results indicated that as the older the students got, the less likely they were to answer with “smaller than the chance of getting tails”, which would indicate a negative recency effect. 35% of the 5th graders, 35% of the 7th graders, and 20% of the 9th graders exhibited the negative recency effect. Only 10% of the 11th graders answered this way, however, and none of the college students did. Fischbein and Schnarch therefore theorized that an individual’s tendency to rely on the representativeness heuristic and other cognitive biases can be overcome with age.[20]

Another possible solution that could be seen as more proactive comes from Roney and Trick, Gestalt psychologists who suggest that the fallacy may be eliminated as a result of grouping. When a future event (ex: a coin toss) is described as part of a sequence, no matter how arbitrarily, a person will automatically consider the event as it relates to the past events, resulting in the gambler’s fallacy. When a person considers every event as independent, however, the fallacy can be greatly reduced.[21]

In their experiment, Roney and Trick told participants that they were betting on either two blocks of six coin tosses, or on two blocks of seven coin tosses. The fourth, fifth, and sixth tosses all had the same outcome, either three heads or three tails. The seventh toss was grouped with either the end of one block, or the beginning of the next block. Participants exhibited the strongest gambler’s fallacy when the seventh trial was part of the first block, directly after the sequence of three heads or tails. Additionally, the researchers pointed out how insidious the fallacy can be—the participants that did not show the gambler’s fallacy showed less confidence in their bets and bet fewer times than the participants who picked “with” the gambler’s fallacy. However, when the seventh trial was grouped with the second block (and was therefore perceived as not being part of a streak), the gambler’s fallacy did not occur.

Roney and Trick argue that a solution to the problems caused by the gambler’s fallacy could be, instead of teaching individuals about the nature of randomness, training people to treat each event as if it is a beginning and not a continuation of previous events. It is their belief that this would prevent people from gambling when they are losing in the vain hope that their chances of winning are due to increase.

 

人們在事件流 E_{t_{-n}}, \ \cdots , \ E_{t_0}, \ \cdots , \ E_{t_{n}} 中尋找『因果』關係的動機強烈。對於當下發生的事想要給出『理由』及『解釋』,故而難以體認事件之『隨機性』與『獨立性』乎。

所以一個賭徒用『手氣』來解釋擲骰子的輸贏不會更自然嗎?除非它是個『造假』的骰子,否則那個人怎麼可能連續丟出十二次六的呢??然而一位剛到場的人,只見到擲了一個六,或以為很普通的吧!此人若被告知已經連續丟出十一次六了,是否會改變他的看法呢!!也許問問求神問卜中樂透者,答案可知也。

終將失之於『大數法則』,落入『小數誤謬』矣!!??

事實上這個『二項分佈』也是通往『常態分佈』的大門︰

正態近似

如果 n 足夠大,那麼分布的偏度就比較小。在這種情況下,如果使用適當的連續性校正,那麼 B(np) 的一個很好的近似是常態分布

 \mathcal{N}(np,\, np(1-p)).

n 越大(至少 20 ),近似越好,當 p 不接近 0 或 1 時更好。[5]不同的經驗法則可以用來決定 n 是否足夠大,以及 p 是否距離 0 或 1 足夠遠:

  • 一個規則是 x=np n(1 − p) 都必須大於  5 。

250px-Binomial_Distribution.svg

n = 6、p = 0.5 時的二項分布以及正態近似

於是人們逐步知道了

中央極限定理

中央極限定理機率論中的一組定理。中央極限定理說明,大量相互獨立的隨機變量,其均值的分布以常態分布極限。這組定理是數理統計學和誤差分析的理論基礎,指出了大量隨機變量之和近似服從常態分布的條件。

歷史

Tijms (2004, p.169) 寫到:

中央極限定理有著有趣的歷史。這個定理的第一版被法國數學家棣莫弗發現,他在 1733 年發表的卓越論文中使用常態分布去估計大量拋擲硬幣出現正面次數的分布。這個超越時代的成果險些被歷史遺忘,所幸著名法國數學家拉普拉斯在 1812 年發表的巨著 Théorie Analytique des Probabilités 中拯救了這個默默無名的理論。 拉普拉斯擴展了棣莫弗的理論,指出二項分布可用常態分布逼近。但同棣莫弗一樣,拉普拉斯的發現在當時並未引起很大反響。直到十九世紀末中央極限定理的重要性才被世人所知。 1901 年,俄國數學家里雅普諾夫用更普通的隨機變量定義中央極限定理並在數學上進行了精確的證明。如今,中央極限定理被認為是(非正式地)機率論中的首席定理。

300px-HistPropOfHeads

本圖描繪了多次拋擲硬幣實驗中出現正面的平均比率,每次實驗均拋擲了大量硬幣。

,深入瞭解了

大數定律

數學統計學中,大數定律又稱大數法則、大數律,是描述相當多次數重複實驗的結果的定律。根據這個定律知道,樣本數量越多 ,則其平均就越趨近期望值

大數定律很重要,因為它「保證」了一些隨機事件的均值的長期穩定性。人們發現,在重複試驗中,隨著試驗次數的增加,事件發生的頻率趨於一個穩定值;人們同時也發現,在對物理量的測量實踐中,測定值的算術平均也具有穩定性。比如,我們向上拋一枚硬幣 ,硬幣落下後哪一面朝上本來是偶然的,但當我們上拋硬幣的次數足夠多後,達到上萬次甚至幾十萬幾百萬次以後,我們就會發現 ,硬幣每一面向上的次數約占總次數的二分之一。偶然必然中包含著必然。

切比雪夫定理的一個特殊情況、辛欽定理伯努利大數定律都概括了這一現象,都稱為大數定律。

400px-Largenumbers.svg

以特定擲單個骰子的過程來展示大數定律。隨著投擲次數的增加,所有結果的均值趨於 3.5(骰子點數的期望值)。不同時候做的這個實驗會在投擲數量較小的時候(左部)會表現出不同的形狀,當數量變得很大(右部)的時候,它們將會非常相似。

從此為『量測數據』之『分析』與『處理』奠定了基礎。

─── 摘自《勇闖新世界︰ W!o《卡夫卡村》變形祭︰感知自然‧數據分析‧五

 

蓋因事件之『隨機性』和機率的『獨立性』難解也。

Independence (probability theory)

In probability theory, two events are independent, statistically independent, or stochastically independent[1] if the occurrence of one does not affect the probability of occurrence of other. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.

The concept of independence extends to dealing with collections of more than two events or random variables, in which case the events are pairwise independent if each pair are independent of each other, and the events are mutually independent if each event is independent of each other combination of events.

Definition

For events

Two events

Two events A and B are independent (often written as  A\perp B or  A\perp \!\!\!\perp B) if their joint probability equals the product of their probabilities:

\mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B).

Why this defines independence is made clear by rewriting with conditional probabilities:

{\displaystyle \mathrm {P} (A\cap B)=\mathrm {P} (A)\mathrm {P} (B)\Leftrightarrow \mathrm {P} (A)={\frac {\mathrm {P} (A)\mathrm {P} (B)}{\mathrm {P} (B)}}={\frac {\mathrm {P} (A\cap B)}{\mathrm {P} (B)}}=\mathrm {P} (A\mid B)}. and similarly
\mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B) \Leftrightarrow \mathrm{P}(B) = \mathrm{P}(B\mid A).

Thus, the occurrence of B does not affect the probability of A, and vice versa. Although the derived expressions may seem more intuitive, they are not the preferred definition, as the conditional probabilities may be undefined if P(A) or P(B) are 0. Furthermore, the preferred definition makes clear by symmetry that when A is independent of B, B is also independent of A.