Meaning of Phred score - probability of base calling error
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.