Mathematical Hadith Verification: A Guide to the New Science of Probabilistic Hadith Transmitter Criticism

Islam’s hadith literature (reports about the actions and sayings of the Prophet ) is one of the most problematic aspects of the religion due to the issues concerning the reliability of transmitters. How do we know if a report going back six or seven generations to the Prophet truly and accurately reports what the Prophet said or did?

So far the science of hadith has largely limited itself to verifying the authenticity of hadiths by verifying the trustworthiness of each person in a hadith’s chain of narrators. If all the transmitters are trustworthy, the assumption is that the hadith is ṣaḥīḥ (“authentic” or “sound”) unless the hadith reports things that are clearly false or contradictory to other hadith narrations. The problem is that a hadith scholar’s own beliefs and biases can strongly affect whether they consider a strange and rare hadith to be authentic or not. A good example is the following hadith in Imām al-Bukhārīʾs collection:

Narrated Abu 'Amir or Abu Malik Al-Ash'ari that he heard the Prophet (ﷺ) saying, "From among my followers there will be some people who will consider illegal sexual intercourse, khazz (a type of clothing), the wearing of silk, the drinking of alcoholic drinks and the use of musical instruments, as lawful. And there will be some people who will stay near the side of a mountain and in the evening their shepherd will come to them with their sheep and ask them for something, but they will say to him, 'Return to us tomorrow.' Allah will destroy them during the night and will let the mountain fall on them, and He will transform the rest of them into monkeys and pigs and they will remain so till the Day of Resurrection."

(Ṣaḥīḥ al-Bukhārī 5590)

A non-scholar who reads this hadith will be greatly troubled by the implication that the use of musical instruments is a characteristic of misguided and impious Muslims. And since the hadith is authentic and present in Ṣaḥīḥ al-Bukhārī, they will face the difficult choice of either believing musical instruments (and hence all music) to be forbidden in Islam, or ignoring the hadith and going with the commonsense and widespread Muslim belief that music is permissible in Islam.

By introducing probability theory into the science of hadith, we gain an extremely powerful tool that enables us to judge just how seriously we should take any particular hadith. This is especially useful in the case of hadiths that seem to be contradicted by other hadiths. For example on the issue of musical instruments we have the following hadith:

Narrated Aisha:

Abu Bakr came to my house while two small Ansari girls were singing beside me the stories of the Ansar concerning the Day of Buath. And they were not singers. Abu Bakr said protestingly, "Musical instruments of Satan in the house of Allah's Messenger (ﷺ) !" It happened on the `Id day and Allah's Messenger (ﷺ) said, "O Abu Bakr! There is an `Id for every nation and this is our `Id."

Sahih al-Bukhari 952

An authentic version of this hadith (Sahih Muslim 892 b) tells us that the girls were using the instrument daff (“tambourine”). So here we have the Prophet approving of the use of musical instruments in his own home, yet the other hadith implies that musical instruments are wicked and unlawful.

Probabilistic hadith criticism helps solve the dilemma of having to choose between two hadiths that are both judged authentic by hadith scholars by telling us which one is stronger.

As it happens, the hadith mentioning the Prophet’s approval of musical instruments is far more “authentic” and believable than the hadith in which he disapproves of them.

In this essay I will use these hadiths on music as an illustration of the probabilistic hadith verification method.

Gathering the hadiths approving of music

The first step in hadith verification is to gather all existing versions of a hadith and their chains and to draw a diagram representing all of its transmitters. Below is a diagram of the version of the “two singing girls” hadiths found in Sahih al-Bukhari:

The blue box is the hadith, and the gray boxes are its transmitters. The first transmitter is Aisha, may God be pleased with her, wife of the Prophet . The second transmitter is Urwa b. al-Zubayr, her nephew. The third transmitter is Urwa’s son Hisham. The the fourth transmitter is the highly respected hadith scholar Hammad b. Usama b. Zayd. The fifth transmitter is Ubayd b. Ismail, a respected hadith transmitter. This transmitter gave the hadith to Imam al-Bukhari who recorded it.

The second version of the hadith is also in Sahih al-Bukhari:

Narrated Aisha:

That once Abu Bakr came to her on the day of `Id-ul-Fitr or `Id ul Adha while the Prophet (ﷺ) was with her and there were two girl singers with her, singing songs of the Ansar about the day of Buath. Abu Bakr said twice. "Musical instrument of Satan!" But the Prophet (ﷺ) said, "Leave them Abu Bakr, for every nation has an `Id (i.e. festival) and this day is our `Id."

Sahih al-Bukhari 3931

Below is a diagram of this hadith’s chain:

You may note that the first three transmitters are the same as those of the previous hadith. So we can join their diagrams into one as follows:

Next is the chain found in Sahih Muslim 892 a:

This chain is exactly the same as the al-Bukhari’s first chain except for the last transmitter. We can therefore join it with the rest as follows:

We now therefore have three chains going back to the same hadith. Imam Muslim mentions two additional chains which we add to the diagram as follows:

We now move on to other collections that mention the same hadith. The first one is a version mentioned in Abu Uwaana’s Mustakhraj, which we add to the diagram as follows:

I have indicated the new chain in green. After further research, I was able to discover a further chain in Sahih al-Bukhari as indicated below:

Al-Bukhari mentions an alternative version of this new chain as indicated below:

The scholar Abu Uwana adds two of his own supporting chains to this version as follows:

We now have a fairly complete picture of all of the different chains of this hadith.

Adding the probabilities

To understand this section, some previous knowledge of probability theory will likely be required.

In order to start, we have to agree on an important assumption. How trustworthy is a single transmitter? This is a matter of intuition. After trying out various probabilities, I have settled on the value of 60%. This means that each trustworthy transmitter in a chain has a 60% chance of truthfully and correctly transmitting the hadith they transmit. This may seem low, but it actually works very well to represent the issues inherent in hadith transmitter verification.

According to probability theory, if you have a witness with 60% probability of truthfulness saying something, and then a second witness also with 60% probability of truthfulness comes along and says the same thing, the probability of both of them saying the truth increases, since we have two witnesses saying the same thing. We calculate this increase by multiplying the falsehood probabilities of the two persons as follows:

Person 1: 60% chance of truth = 40% chance of falsehood

Person 2: 60% chance of truth = 40% chance of falsehood

The chance of falsehood for both = 40% * 40% = 16%

The chance of truth of both = 100% - the chance of falsehood = 100% - 16% = 84%

Below is a diagram of an imaginary chain that represents these facts:

We have two people each of whom say that the Prophet said something. Each of them has a 60% probability of truthfulness and accuracy, but the two of them together have a 84% probability of truthfulness and accuracy, so the above imaginary hadith has an 84% chance of truthfulness and accuracy.

As we add more transmitters, the probability continually goes up:

Now we have four supporting witnesses, so their probability of truthfulness can be calculated as follows:

Falsehood probabilities: 40% * 40% * 40% * 40% = 2.56%

Truth probability: 100% - 2.56% = 97.44%

What this means is that if you have four Companions transmit the same hadith from the Prophet , the chance of the hadith being truly and correctly transmitted is 97.44%, which is a very high chance.

In probability theory 1 represents 100% and 0 represents 0. So another way of doing the above calculation is as follows:

0.4 * 0.4 * 0.4 * 0.4 = 0.0256

1 - 0.0256 = 0.9744

The benefits of the 60% assumption will become clear as we go through the verification process of the chains we gathered earlier. We will first deal with the top part of the diagram, which is as follows:

First, we will fill in the original 60% assumption by writing 0.6 in each bubble:

We do not add the 0.6 to the bubbles on the far right because those are written books, so they virtually have a 100% chance of reliability. What concerns us are unwritten, oral transmissions. The first step is to combine the three middle transmitters’ probabilities vertically, as follows:

Falsehood chances: (1-0.6) * (1-0.6) * (1-0.6) = 0.064

Truth chance: 1 - 0.064 = 0.936

Above, each of the three transmitters has a 100% – 60%, or 40% likelihood of falsely transmitting the hadith. But in order to get the probability of all of them falsely transmitting the hadith at the same time, we have to multiply these chances together: 40% * 40% * 40% = 6.4%. So the chance of all of them having falsely transmitted the hadith is only 6.4%, meaning that there is a 93.6% chance of them having truly and accurately transmitted this hadith.

We will call this “vertical combination”. We vertically combine all of the probabilities of the truthfulness and accuracy of the transmitters to get a single number, in this case 93.6%, which represents the authenticity of all them combined together.

We next have to do a horizontal combination. All three transmitters transmit the hadith from Hammad b. Usama, who also has a 0.6 (60%) chance of authenticity.

In the case of horizontal combination, rather than multiplying falsehood probabilities, we multiply truth probabilities. The reason is that as information passes down through a chain of transmitters each of whom have a 60% chance of authenticity, the chance of the information being correctly passed down decreases. There is more chance for error and fabrication. If you hear a Companion say that the Prophet said something, that is far more trustworthy than another person saying they heard their father say that their grandfather said that a Companion said that the Prophet said that.

So we multiply 0.93 by 0.6 to get 0.558, which means 55.8%. We now update the bubble for Hammad b. Usama to represent this new probability:

Hammad b. Usama’s probability went down from 0.6 to 0.558 because we did not hear the hadith directly from him, but from three people who claim to have heard him. Since those three people have a combined probability of 93% truthfulness and accuracy (rather than 100%), this slightly decreases the probability of the truthfulness and accuracy of the information we get from Hammad b. Usama.

We now move on to al-Bukhari’s second chain, which is as follows:

Using horizontal combination:

0.6 * 0.6 * 0.6 = 0.216

So this second chain has only a 21.6% probability of truthfulness and accuracy. The reason is that we do not have any supporting transmitters. We only have Muhammad b. al-Muthanna’s word for it that Muhammad b. Jaafar said that, and we only have Muhammad b. Jaafar’s word for it that Shu`ba said that. The deeper a chain goes back in time, the lower its probability of authenticity and accuracy becomes.

We next deal with two chains from Sahih Muslim:

Using vertical combination for the middle transmitters, we get a probability of 0.84 or 84%. Multiplying that by 0.6, we get 0.504, meaning that Abu Mu`awiyah has a 50.4% probability of authenticity.

Next we have the more interesting task of combining all the chains we examined above. We have to vertically combine the probabilities of Hammad (0.558), Shu`ba (0.216) and Abu Mu`awiyah (0.504):

Falsehood probabilities: (1-0.558) * (1-0.216) * (1-0.504) = 0.171877888

Truth probability: 1 - 0.171877888 = 0.828122112

So the probability of the truth of the information coming from these three transmitters is 82.8%.

Now we horizontally combine this with Hisham b. Urwa’s 0.6 probability, resulting in 0.496873267

We repeat these same steps for the remaining chains, as follows:

And by vertically combining the left-most probabilities (Hisham b. Urwa’s 49.68% and Muhammad b. Abd al-Rahman’s 21.04%), we get 0.602766552, so the information has a 60.27% probability of authenticity at this stage.

We next multiply this 0.602766552 by Urwa b. al-Zubayr’s 0.6 (horizontal combination), resulting in 0.361659931.

In the final step, we multiply this result by Aisha’s 0.6:

The result is 0.216995959. This means that the hadith with all of its chains has a 21.69% chance of authenticity and accuracy. This seems very low, but it is natural for a hadith that comes from a single Companion, through a single transmitter.

In my methodology, a hadith that reaches 20% or higher is ḥasan, a hadith that reaches 30% or higher is ṣaḥīḥ, a hadith that reaches 60% or higher is ṣaḥīḥ al-ṣaḥīḥ (a degree above ṣaḥīḥ), and a hadith that reaches 85% or higher is mutawātir (“widely transmitted”, a degree above ṣaḥīḥ al-ṣaḥīḥ).

Different scholars may prefer different probabilities for what they consider ḥasan or higher. Personally these are the numbers I have settled on that my heart is comfortable with and accepts.

The hadith disapproving of music

We now move on to Ṣaḥīḥ al-Bukhārī 5590, the hadith that says misguided Muslims will approve of musical instruments. Below is a diagram of the hadith’s chain:

The hadith comes from a single chain that does not branch out, meaning it is an extremely questionable chain, which I call a “precarious” chain. To calculate its probability of authenticity, we horizontally combine all the 0.6 probabilities as follows:

Probability of truth = 0.6 * 0.6 * 0.6 *0.6 *0.6 *0.6 = 0.046656

This hadith therefore has a 4.66% chance of authenticity, which is extremely low. In my methodology a hadith between 10% and 20% is munkar (strange and likely to be false), while a hadith below 10% is ḍaʿīf (“weak”, i.e. almost certainly false). As discussed in another article, there is another reason to consider this hadith false: it says misguided Muslims will consider khazz (a type of clothing) lawful. But we know 20 Companions of the Prophet wore this type of clothing, so this makes the hadith even more dubious.

The verdict

So this hadith in which the Prophet says misguided Muslims will consider musical instruments lawful has only a 4.66% chance of authenticity, while the hadith in which the Prophet approves of musical instruments has a 21.69% chance of authenticity. Thus the hadith in which the Prophet approves of musical instruments is 4.6 times stronger than the hadith in which he disapproves of them (21.69 ÷ 4.66 = 4.65).

Based on this, we can confidently say that the evidence in support of the lawfulness of musical instruments is far stronger. Therefore approval of musical instruments is a vastly better representation of the Prophet’s sunna (tradition) than disapproval of them.

Defending the assumptions

The assumption that each transmitter has a 60% chance of accurately and truthfully transmitting the information needs to be defended. This assumption makes most ṣaḥīḥ narrations get an authenticity around 30%. If we treat these numbers empirically, it would mean that most authentic narrations are more likely to be false than true. For this reason I have chosen to treat all hadiths that reach 30% or higher as ṣaḥīḥ / authentic despite the fact that empirically they fall beneath the truth threshold of 50%.

My reasoning is that by choosing these assumptions, we create an interface between the empirical research world and the classical hadith studies world. A non-Muslim scholar may choose the 30% verdict on an authentic narration as a cause for skepticism, while a Muslim scholar can continue to make use of probability theory while adopting the traditional view of considering individual, low-truth-probability hadiths as authentic.

Another reason for the 60% assumption is that it gives us a very wide range of results. Some hadiths will reach close to 100%, many others will hover between 50% and 20%. This allows for representing a scholar’s empirical intuitions about these transmitted pieces of information, while also allowing them to remain in the classical hadith criticism world if they wish by choosing lower numbers, such as 30%, to represent full authenticity.

The study of Islam works on the basis of the autonomous consensus of the researchers, a concept I have defended elsewhere.1 Therefore introducing probability theory into hadith criticism will require the involvement of many researchers until a consensus emerges on the best assumptions to be chosen. Therefore my assumptions in this article are merely preliminary suggestions intended to illustrate what a probabilistic study of hadith would look like.


Combining probability theory and hadith criticism can greatly help in resolving issues surrounding contradictory hadith narrations by making it clear which hadiths are superior to which ones. It would be extremely helpful if we could build a new hadith collection that shows the probability of authenticity of each hadith rather than merely saying whether it is authentic or not. But such a work would require many years of work.

There is also the issue of sub-par transmitters. While I have chosen a probability 60% for reliable transmitters, for transmitters who are mudallis, non-qawī, non-ḥujja, or considered weak by some scholars and not others, lower probabilities will have to be used where necessary.

There are also different levels of weak transmitters. Some are considered weak for ideological reasons (because they held beliefs that hadith scholars considered unacceptable), while others are considered weak because they were caught lying or fabricating hadiths. Different probabilities will have to be used for different levels of weakness.

Further applications of the probabilistic method

Below is a list of studies I have conducted on certain topics of hadith criticism where I use the methodology developed in this article:

Ikram Hawramani, “Islam and magic”, The Hawramani Institute,, Apr 02, 2019, retrieved April 22, 2019.

Ikram Hawramani, “Is ruqya part of Islam? A study of the hadiths on ruqya using probability theory”, The Hawramani Institute,, Mar 24, 2019, retrieved April 22, 2019.

Ikram Hawramani, “A study of “the majority of people in Hell are women” hadith”, The Hawramani Institute,, Mar 26, 2019, retrieved April 22, 2019.


  1. See Ikram Hawramani, “Solving the Problem of the Codification of the Sharia”, The Hawramani Institute,, Apr 16, 2019, retrieved April 22, 2019.

2 thoughts on “Mathematical Hadith Verification: A Guide to the New Science of Probabilistic Hadith Transmitter Criticism

    1. Ikram Hawramani Post author

      There are different levels of “weak”. Some narrators are considered weak by a few scholars and reliable by others, while there are also those who are considered weak by most of the scholars. A weak narrator can possibly be a given a probability between 0% to 20% depending on just how weak they are.


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