Let's Talk about I Origins and Central Limited Theorem

[Spoiler Alert] This article spoil the I Origins straight down to the end. If you want to watch this film but haven't, I would recommend you to finish the film first. 

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  In the I Origins, there is a scene that Grey visited India and met Priya. He made Priya do a sequence of tests, which is picking a prefer one from three pictures for 25 sets. As a result, Priya made a 44% accuracy compare to the result that Grey thought Sofi would choose. Obviously, this result is above the average. But could we say that it is convincible? And how convinsing it is? Here comes the Central Limited Theorem to rescue.

  For those who has no prior knowledge about Central Limited Theorem. The theorem is a easier way to estimate how likely for a lot of I.I.D. events occurred less than a given times. (For those who don't really know statistics, you can treat I.I.D. as a lot of repeated experiments which, of course, have the same distribution.)

  In mathematical representation, it is $$P(S_n \leq k) \approx \phi(z); z= \frac{k - n}{\sigma \sqrt{n}}$$

which means the probability of the sum of $n$ times I.I.D. outcomes are lesser than or equal to $k$ is close to $\phi(z)$, where $\phi(.)$ is the CDF(cumulative distribution function) of normal distribution, $\sigma$ and $\mu$ are the standard deviation and mean of the distribution of the given event respectively.

  And now, we are ready for some applications. In the film, Priya have done 25 tests and got a 44% accuracy, which simply indicates Priya got 11 true answers. We assume that Priya had guessed randomly under uniform distributed. Therefore Priya had a 33.3% probability of true, and 66.7% for false. This is a Bernoulli distribution, so we got a $\mu=\frac{1}{3}$ and $\sigma^2=\frac{2}{9}$. Priya did 11, so let us estimate how likely a person can achieve 11 or more in totally random selection.

  That is $P(S_{25} \geq 11) = 1 - P(S_{25} \leq 10)$, and we knew the following by Central Limited Theorem.
$$P(S_{25} \leq 10) \approx \phi(\frac{10-25*\frac{1}{3}}{\sqrt{25*\frac{2}{9}}})=\phi(\sqrt{2}) \approx \phi(0.707) \approx 0.7794$$

To interpret, $P(S_{25} \leq 10)$ is the probability of a guy answered 10 or less correctly in 25 tests, and as we can see, about 78% people are able to do so. On the other hand, $P(S_{25} \geq 11)$ is the probability of a person reach 11 or more in the 25 tests, 22% people can do so. That is about 1 out of 4~5 people would reach such an accuracy.

  I guess that is not promising enough to show that Priya inherited Sofi's soul, is it? Not to mention that some of those pictures are beauty and naturally attractive, which increases the probability of Priya made a choice that same as Sofi does.

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  I Origins elaborated that a scientist(represented as ration) still a human, who is emotional more or less. Grey knew that Sofi may not be a good mate, but he enjoys her love. That is the difference between lover and mate. Grey's passion about iris and the misrecognized result from the recognition system are the last straws, which push him to find out if extremely similar irises connect to the same door of a soul. And the irrational of Grey made a fortune for both him and Priya. Although this is not part of the story, I hope Grey adopt Priya after that and bring her a better life.

  It is not an amusement film, but a film can make us think more after watched it.