In Tensorflow, What kind of neural network should I use?
Sure, you can use an RNN. I would create two features for the past $k$ run lengths, as well as the length of the current run; e.g., just before t=150, the current run would be length 2 (red), and previous three runs would be (1,1,1) for red and (1,1,5) for blue. The intuition here is that the run lengths seem to follow some sort of exponential distribution, and you want to help the model estimate the scale parameter by feeding it samples of the length. You could additionally encode the past k events as a bit string, with 1 representing red and 0 representing blue. You have a classification problem, so you should use classification loss like the cross-entropy, and a softmax output layer to get your probabilities.