If a prime number P(n) takes the form x mod 3 (where x is either 1 or 2), is the probability that P(n+1) takes the same form = 0.50? All prime numbers greater than and equal to 5 take the form 1 mod 3 or 2 mod 3. Furthermore, there are equal numbers of each in the number line, stretching on to infinity.
Given these facts, it would seem reasonable to assume that if P(n) were, say, of the form 1 mod 2, then there would be a 50:50 chance that p(n+1) would also take this form.
An analogy could be, say, tossing a coin. Having tossed a 'head' it is just as likely that the next toss will also be 'heads' as a 'tail'.
An analysis of all primes from 5 to 1.6 Billion ( drawing on work by S Ares and M Castro ), however, reveals that this is far from the case for the first stages of the number line.
I calculate that the number of occasions where consecutive primes both take the form of either 1 mod 3 or 2 mod 3 is 35,447,544
whereas the number of times they differ ( 1 mod 3 followed by 2 mod 3 or visa versa ) is 43,995,257
This is a difference of some 8,547,713 primes or around 10.8% of the total.
a quick count of the 500 primes between 999,999,982,843 and 999,999,997,391 reveals a similar discrepancy:
Same form: ( 1 mod 3 followed by 1 mod 3 or 2 mod 3 followed by 2 mod 3 ) = 219 times
diff form  ( 1 mod 3 followed by 2 mod 3 or 2 mod 3 followed by 1 mod 3)  = 281 times
in this case a difference of  62 or 12.4%
Clearly, if my conjecture is true, the point at which these two totals converge, must be truly astronomical.
 A: The answer (in the limit) appears to be yes, but in practice (taking numbers up to any finite large number) the answer appears to be no, and in a predictable way. This is the primary idea of recent work of Robert Lemke Oliver and Kannan Soundararajan. Specifically, they conjecture that there is a secondary term in the asymptotics that explain the biases one can observe in practice.
In their paper they explicitly consider the possibilities mod $3$ in their introduction. For example, among the first million primes, they note that there are approximately $215000$ primes congruent to either $1$ or $2$ mod $3$, and whose next prime is also congruent to $1$ or $2$ mod $3$ (respectively --- and by $21500$ I mean $215000$ each for the $(1, 1)$ pairs and the $(2, 2)$ pairs), while there are about $285000$ (each) for the $(1, 2)$ and $(2, 1)$ pairs.
Generically their conjecture is of the shape that prime ensembles containing primes with the same residue class should occur somewhat less often (in secondary order terms). For the rest, I defer to the paper.

Reference
Unexpected biases in the distribution of consecutive primes, by
Robert J. Lemke Oliver, Kannan Soundararajan, https://arxiv.org/abs/1603.03720
