Below is a problem that we find hard to solve, any input would be much appreciated. In short, we're looking for the exact probability after n tries, and the corresponding formula for this problem.
Problem description Alice and Bob both have their own vase, and in each vase there is a red, a blue, a green and an orange marble (2 vases, 8 marbles). Bob and Alice simultaneously pull out a marble from their own vase, note the result (either red, blue, green or orange), and put their marble back in their own vase. Bob, a true gentleman, gives Alice a head start of +1 for all colors, so the initial score is Bob: 0, 0, 0, 0, and Alice: 1, 1, 1, 1.
Bob wins when any of the scores for Bob turns +1. For example, the score for Bob is: 4, 4, 4, 4, and Alice: 3, 6, 6, 5. Note that only one color should be +1, and Bob wins.
The first question: What is the chance that Bob will overtake (+1) Alice on any of the colored marbles after n tries? And how to express this is in formula?
The second question: What is the chance that Bob overtakes Alice on any of the colored marbles after n tries, where: 1. The speed in which they draw differ (e.g. Alice draws p-times faster than Bob), and, 2. Bob provides Alice with a greater head start (e.g. initial score: 0,0,0,0 vs 2,2,2,2) And how to express this in a formula?
Solution directions So far, we've tried bi- and multinomial distributions, but these don't seem to solve the problem for putting it into a formula. Simulating the problem is easy in R, but we're looking for the actual probabilities and formula. Our current idea is to use 'countably infinite Markov chains'.
Any (other/supplemental) feedback are welcome!
Thanks for reading.