Red vs Black

A bag contains 3 red and 4 black balls. Abhineet, randomly decides to toss a coin or a die with probabilities \( \frac{1}{3} \) and \( \frac{2}{3} \) respectively.

If a die is tossed, and the number obtained is a multiple of 3, he does not remove nor add extra balls into the bag. If the number is not a multiple of 3, he picks a ball randomly out of the bag, and replaces the ball along with an additional ball of the same colour.

If a coin is tossed, and a head is obtained he picks a ball randomly of the bag and discards it. If a tail is obtained, he discards one of the black balls.

He does this process once.

After the process is done, he picks a ball out of the bag randomly.

Given that the probability of picking a red ball is \( \dfrac{A}{B} \), where \(A\) and \(B\) are coprime positive integers.

Find \( B - A \).


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