2 defending dice do battle with a attacking dice, where a can be any positive integer. The highest scoring defender die is paired with the highest scoring attacking die; the second-highest scoring defending die is paired with the second-highest scoring attacking die. If, in each pair, the defending die has a greater or equal score to the attacking die, the defending die wins (conversely, if the attacking die has a greater score than the defending die, the attacker wins that pair).
In terms of a, how do we calculate the probability that i) the defending dice will win both pairs, ii) the attacking dice will win both pairs, iii) 1 pair will be won by a defending die and the other by an attacking die?
I see I'm required to show work, but I really don't know how to start! I solved the problem for when there is only 1 defending die and it's paired up with the highest attacking die. I landed up realizing that P(Defending die wins pair)=(1/6)*Sigma(i=0 to 6)[(i/6)^a]. (Please excuse the lack of LaTeX - my abilities do not reach far enough so far to write sigma functions in LaTeX) but how do I find a function for this problem now there are 2 defending die?
In terms of a, how do we calculate the probability that i) the defending dice will win both pairs, ii) the attacking dice will win both pairs, iii) 1 pair will be won by a defending die and the other by an attacking die?
I see I'm required to show work, but I really don't know how to start! I solved the problem for when there is only 1 defending die and it's paired up with the highest attacking die. I landed up realizing that P(Defending die wins pair)=(1/6)*Sigma(i=0 to 6)[(i/6)^a]. (Please excuse the lack of LaTeX - my abilities do not reach far enough so far to write sigma functions in LaTeX) but how do I find a function for this problem now there are 2 defending die?