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I learned that Phred quality scores are logarithmically linked to error probabilities. For example, if Phred assigns a quality score of 10 to a base, the chances that this base is called incorrectly are 1 in 10. How is it possible to have 10 possiblities when you can only have 4 bases (ACGT) ? When assigning a base, there are only three ways to be wrong to me.

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Phred score is a measure of the probability that a base call is wrong/right.

Your understanding of this probability as "if Phred assigns a quality score of 10 to a base, the chances that this base is called incorrectly are 1 in 10", is not entirely accurate.

Quality scores calculation and base calling are specific to the machine. The way an Illumina HiSeq calls a base is different from how PacBio does it.

For example, Illumina uses high precision cameras to record the colors generated during sequencing (each color represents a base). Moreover, it does not sequence a single molecule of DNA but a cluster. Now, the machine's base calling software will calculate the reliability of the light signal being interpreted as a base (let's say A). Several parameters are considered in this calculation.

For each base call, a number of quality predictor values are computed. Quality predictor values are observable properties of clusters from which base calls are extracted. These include properties, such as intensity profiles and signal-to-noise ratios, measure various aspects of base call reliability. They have been empirically determined to correlate with the quality of the base call.


Illumina Technical notes: Understanding Illumina Quality Scores

Likewise, different machines have different techniques for reading a base signal and computational models to calculate the accuracy of the signal.

How is it possible to have 10 possiblities when you can only have 4 bases (ACGT) ?

Why not? You are just calculating what is the probability that a base is not adenine. The minimum probability should be more than 1/4, at or below which the base call wouldn't probably even assign the base as adenine.

Imagine it like this: if adenine is represented by blue light and you see that in the spot there is not a pure blue signal but some interference of other colors (noise) which represent 10% of the signal. There can be other parameters too that make the calculated blue signal to deviate from the "ideal signal". The percentage of non-overlap with the perfect signal would be the error.

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if Phred assigns a quality score of 10 to a base, the chances that this base is called incorrectly are 1 in 10

This does not mean that there are 10 different possibilities for a base. This means that you have 10% chance that the base is different from the one that was called.

If the base called is an A with a score of 10, it has 90% chance of really being an A, and 10% chance of being either C, G or T. If we assume that the probabilities of each of the 3 other bases are equal, this would be:

             |   A   |   C   |   G   |   T   |
-------------+-------+-------+-------+-------+
P(real_base) |  0.9  | 0.033 | 0.033 | 0.033 |

However, I'm not sure that the probabilities of C, G and T are really equal: it could well be that, due to the specifics of the sequencing technology, some (read base, real base) pairs are more common than others.

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