The fallacy of Backward Probability Calculation

Picture of Heath Ledger playing as the Joker character in The Dark Knight (2008) movie. He is holding a Joker card in his right hand while looking into the camera with a red smile carved on his face as in the movie.

By Eric Bright

Here is a simple puzzle for you with deep implications.

I have a deck of Bicycle cards with 52 cards plus two ? Jokers (a black and white and a coloured one), as well as one Bicycle ? introduction card, and an advertising card (56 cards in total).

I have been shuffling them for the past two months or so. Today, I got the following sequence of cards:

K♥, 5♠, 6♦, 5♥, 8♠, 7♦, K♣, 8♣, A♠, 4♥, 2♥, J♥, 8♦, ?C, 3♠, Q♦, ?B&W, A♥, 5♦, A♦, 9♠, Q♣, 2♣, 10♣, 3♦, K♠, J♦, 7♥, ?-ad., 9♦, 7♣, A♣, 3♥, J♣, 8♥, 4♣, 3♣, 4♦, 2♠, 10♠, ?-intro, Q♠, 9♣, 6♣, 10♥, 7♠, J♠, 4♠, 6♠, 5♣, 6♥, 10♦, 9♥, Q♥, K♦, 2♦.

Question 1

Can you calculate how probable it is for anyone to get this exact sequence after a dozen random shuffles?

Once you are done, then I will have another question for you if you can successfully answer that one.

The answer to Q1

The answer is 1/(56!).

That is equal to:


Which is almost:


Astronomically low chances, eh?

Question 2

Despite the very small chances of anyone getting that very sequence of cards, I’ve got it already. How come?

What is wrong?

Well, something doesn’t seem right here, don’t you think so?

The problem with this question is that we are calculating the probability of a given event after the fact, i.e. backward.

The event has already happened. So, its actual probability of happening is 1, or 100%. Now, if you want to play a trick on your audience, or, to be more technical, if you want to use a fallacy to prove your point, then you can try to distract the audience and calculate the probability of the same event, that has already happened, as if something hypothetical that might happen in the future randomly. That way, you will get an absurdly-unlikely probability, which is mathematically correct.

So, if a person is going to shuffle the same deck of cards and deal random sequences of cards, they will have a very low chance of getting a hand that is exactly like the hand above. Their chances are 1.406 × 10-75. That is unimaginably low.

If they keep getting hands after hands every single second, they will not get a hand like mine starting from the beginning of the universe till now (the universe is almost 4.35… × 1017 seconds old). Not even close. They will need to keep dealing hands a few more times, a bit longer than the current age of the universe. How much longer?

1633740385510333375277680459653483090262837362042111600000 times longer.

That is:

(number of times they need to deal cards to get my sequence)/(age of the universe in seconds) =
(56!)/(age of the universe in seconds) =
(56!)/(4.351968 × 1017)
≈ 1633740385510333375277680459653483090262837362042111657731.5 times the age of the universe.

For all practical purposes, that is close enough to being practically, totally impossible. We can confidently say that it will not happen.

And yet, it happened for me.

So, am I lucky or what?

What is the point?

“It has already happened” is completely different from “how likely it is to happen again.” And that is the point. One has the probability of 100%, and the other has the probability of
1.406 × 10-75. One is absolutely possible, and the other is absolutely impossible.

One is absolutely possible, and the other is absolutely impossible.

What are the implications?

Any time I talk to a layperson regarding religion, the conversation comes down to this point. I am going to call it the Fallacy of Backward Probability Calculation.

The fallacy of backward probability calculation happens when someone claims that an event that has already happened (p=1), is impossible to happen (p≈0), because the probability of an exact event to randomly happen again is almost zero.

So in this fallacy, they are comparing the probability of an event that has already occurred (randomly or otherwise), with the probability of a hypothetical, identical event in the future should it happen randomly. Then assigning the probability of the latter to the former.

When this fallacy is applied to life and evolutionary biology, the results can be mind-boggling. People who commit this fallacy in their reasoning usually have no idea that they are doing so. It’s “obvious” to them that the event in question is virtually impossible to happen randomly, and it still happened. “Therefore,” they would say, “there has to be a guiding hand behind the scene to direct the universe to generate this impossible event.” And that “hand” has to be the hand of their particular brand of god or gods.

To their minds, life is almost impossible to happen, and since it has happened already, their god must have made an impossible thing to occur.

The same fallacy was deployed in Kitzmiller v. Dover Area School District case, in 2005 in the US. It is being used in countless conversations between people every day all over the world. It is one of the most perplexing enigmas in the minds of religious people, over which they cannot get.

One does not have to have a mathematics or statistics degree to understand the fallacy once it is explained. However, it is rarely explained at all in conversations. When conversations come to this point, both opponent and proponents of Intelligent Design, ID, have a hard time to see through the faulty reasoning.

I blame the opponents of ID for not spending enough time to understand the fallacy well enough so they can quickly call it out and effectively explain it away with good examples and demonstrations.

The above-mentioned puzzle can be used to draw attention to the fallacy and its implications. That puzzle can be touching and tangible to most audiences. It can be discussed and demonstrated in parties, gatherings, and almost any discussions and debates with the help of a simple deck of cards. That can easily bring the point home to anyone who is paying attention. It is simple, short, and much less confusing than the concepts such as DNA and other complex life structures.

The other problem with this fallacy is that it assumes that evolution is random. The elimination of the less fit genes are hardly a random event. It almost always follows recognizable patterns and certainly follows the laws of nature. The game of life is far from being random.

The “random” concept comes into the scene only when mutations happen (by the way, technically, mutations are not random either. They happen following the same laws of nature. But, to make things easier to calculate and follow, we assume that most gene mutations happen randomly, which is not the truth anyway, but a gross approximation). Everything else in the evolution of life is totally non-random. Nonetheless, this misinformation and the answers to it won’t get a chance to even come up in the same conversations when the whole time is spent on being mesmerized by the “miracle” of complex life occurring despite its “improbabilities.” The probability issue, therefore, is a more urgent fallacy in this circumstance. Then, if any time is left, one can address the apparent role of “randomness” in evolution.

At any rate, I invite my kind readers to discuss this newly-coined fallacy in public settings anytime that it comes up. Raising awareness of the issues in the process of reasoning that goes underneath the surface of this fallacy might help more and more people to see it for what it actually is: a fallacy.

I called it the fallacy of backward probability calculation, or BPC for short, but please feel free to come up with more intuitive names for it and let me know. I will make sure to include your suggestion in the article with proper credit.

Happy New Year!

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