Don't know a little statistics, "Star Wars" for nothing

A long time ago , In a distant galaxy ……

Our hero · Solo is driving “ Millennium Falcon ” No. 1 dodged the pursuit of Star Destroyers and titanium fighters at full speed .

In a flurry , robot C-3PO The tip says ,“ Millennium Falcon ” The superluminal thruster of No. 1 is broken . What's worse is , The spacecraft was also hit by an asteroid .

This hit Han · Soro inspiration . I saw him sitting in the driver's seat , Adjustment status , take “ Millennium Falcon ” The No. 1 went straight into the asteroid belt .

Always knowledgeable C-3PO Hurriedly prompted behind him ：“ chief , The probability of successfully crossing the asteroid belt is about 1 Than 3720！”

“ Never mention probability to me ！” han · Solo replied .

《 Star Wars: ： Empire counterattack 》 Still

（ source ：imdb.com）

If seen 《 Star Wars: ： Empire counterattack 》, Then you must have fresh memories of the above scenes . Behind the gripping plot , In fact, there is an interesting statistical knowledge hidden . Let's take a closer look at .

One side , We all know Han who has the aura of the protagonist · Solodine can successfully cross the asteroid belt . On the other hand , We also know that C-3PO The data is reasonable , Because no titanium fighter has successfully crossed the asteroid belt .

han · Solo thinks he has the probability of successfully crossing the asteroid belt = 100%

C-3PO The calculated probability of successfully crossing the asteroid belt = 1 / 3720 = 0.027%

Since both sides make sense , Then why is the difference between the two probabilities so large ？

likelihood ：C-3PO The way of reasoning

C-3PO There's nothing wrong , It just forgot the important information —— han · Solo is a master , Super powerful ！

C-3PO There is no nonsense , It is proficient 600 More than 10000 communication modes , This requires a lot of data support . So we can basically determine ,“1 Than 3720” The statement of is supported by data .

We according to the C-3PO Data given , Use β Distribution to determine the probability of success . Why β Distribution , Because given the success and failure ratio , It can correctly simulate various possible probabilities of an event .

β The parameters of the distribution are α（ The number of successes observed ） and β（ Number of failures observed ）：

Beta(α,β)

This distribution tells us , According to the current data ,C-3PO What is the most likely probability of successfully crossing the asteroid belt .

To find out C-3PO The way of reasoning , Here are some reasonable assumptions . We assume that C-3PO Only two people have successfully crossed the asteroid belt , While the rest of the 7440 People ended their journey to the asteroid belt in a violent explosion .

The following figure shows β Distribution , namely Beta(2, 7440).

The hope of success is so slim ！ No wonder Lian Han · Solo himself thinks it's dangerous to rush into the asteroid belt , So he said ：“ They must be crazy to follow us .”

transcendental ： han · The power of solo

C-3PO The problem with the analysis of is , It has data about all pilots , Dan Han · Solo is not an ordinary pilot .

Our transcendental belief is , han · Solo is sure to successfully cross the asteroid belt , Because he has the aura of the protagonist .

The probability of a priori belief is often controversial . A lot of people think “ Fabricate ” A priori probability is not objective . But you need to know , It is more absurd to deny a priori probability .

We have many reasons to believe that Han · Solo will survive , But there is no data to support this belief . First, let's see the upper bound of his severity .（ Yes , There should be an upper bound , isn't it? ？ If we believe that he will never die , Then this movie is old-fashioned and boring .）

We have a strong belief that Han · Solo will successfully cross the asteroid belt . Here again, make reasonable assumptions . Suppose our belief that he will survive is 20000 Than 1.

The figure below shows that we believe in Han · A priori probability distribution that solo will survive , namely Beta(20000, 1).

Posttest ： How writers create suspense

We have made sure that C-3PO Belief （ likelihood ）, Also established the Han · Thoreau's belief model （ A priori probability distribution ）. Next, we need a way to combine the two , Form the final posterior belief .

Let's combine the above two as follows β Distribution , A posteriori probability distribution is obtained .

Insert specific values , Yes ：

Beta(2+20000,7440+1) = Beta(20002,7441)

The posterior probability distribution can be seen intuitively from the figure below .

You can see , By way of C-3PO Belief and “ han · Solo is great ” Combine your beliefs , We found that the probability of his successful survival is about 73%.

This means that we still think Han · Solo has a good chance of success , But we will still worry about him .

Conclusion

Through this example , We learned that likelihood 、 The relationship between a priori and a posteriori . actually , These three are Three elements of Bayesian theorem .

C-3PO The data provided with us on Han · Thoreau's cognitive ability does not match the likelihood , But we did not directly deny C-3PO The data of , Instead, we combine likelihood with our prior beliefs , Form the final belief after adjustment （ Posttest ）.

In the movie , This uncertainty is crucial to creating tension .

in real life , Master the basic logic of Bayes theorem , You can also use it to analyze any uncertain situation you encounter .

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