The probability that abc ab a is divisible by 3 is 13/27.
Probability
Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.
We group this into groups of 3, because 3|2010. This means that every residue class mod 3 has an equal probability.
If 3|a, we are done. There is a probability of 1/3 that that happens.
Otherwise, we have 3 | bc + b + 1, which means that b (c + 1 ) = 2 mod{3}. So either
b = 1 mod (3) c = 1 mod (3)
or
b = 2 mod (3) c = 0 mod (3)
Which will lead to the property being true. There is a 1/3 * 1/3 = 1/9 chance for each bundle of cases to be true.
Thus, the total for the cases is 2/9. But we have to multiply by 2/3 because this only happens with a 2/3 chance. So the total is actually 4/27.
The grand total is
1/3 + 4/27 = 13/27
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