Another "lottery-like" mathematical challenge...

We have an experiment with a professor and student. The professor has a container that has a very large number of balls, both red and blue in color, with hundreds or more of each. The balls within each color are identical and indistinguishable. In front of him he has two empty jars. Let's call them Jar 1 and Jar 2. The experiment will be done out of sight of the student, who will only see the final result, and will then be presented with a question. The professor starts the experiment with Jar 1. He flips a fair coin. If the coin is Heads, he places a red ball in Jar 1, else if Tails he places a blue ball. If the selected ball is red, he flips the coin again and picks another ball per the stated rule. If the ball is red each time, he continues to sample until he reaches a maximum of 5 balls, then stops and moves on to Jar 2. If any selected ball is blue, he stops sampling any more balls for Jar 1 at that point and moves on to Jar 2. For Jar 2 the setup is the same except that instead of a coin, the professor uses a fair six-sided die. If 1 or 6 is rolled the ball selected is blue. Otherwise, red is selected. Again, this continues until a maximum of 5 red balls is reached or a blue ball is drawn (in which case no more balls are drawn).

Finally, the professor flips a fair coin again, with Heads representing Jar 1 and Tails Jar 2. Within the selected Jar, he picks one ball at random (with the possibility that there may only be one blue ball in the jar).

The professor finally reveals to the student that the selected ball is blue. The jars and starting container are still out of sight of the student. The student is explained the procedure and rules and is asked what the probability is that the selected blue ball came from Jar 1. What is the solution?