A and B are playing " board football", a two player in which the objective is to score as many goals as possible. As the game does not have any terminating statement, an infinite number of rounds are played.(Similar to having an infinitely large board of Scrabble with infinite number of tiles,for example www.babbl.xyz)
The rules are as follows:
1)The player with the ball(say A) has to make a certain number of "passes" before they get to shoot. The player in possession (here A) has to roll an n-sided dice, and the number that turns up is the number of passes they make within that round, unless they are intercepted.
2)If the other player/the defender(say B) rolls the dice and gets the same number, then an 'interception' takes place and now B is in possession of the ball and the game continues. (Note that the a round is said to have been completed when the attacker and the defender both have rolled the dice(s).)
3)If the player in possession manages to complete/achieve the required number of passes, they have the opportunity to shoot. This can be done in multiple ways. For example, if the number of required passes are 8 and one six sided dice is being used, then the attacker can reach this number by rolling 2-3-2-1 , 6-2 , 1-1-1-1-1-1-1-1 and so on.
4)The number of passes made by the player in possession can be higher than the required number of passes.For example if the number of required passes are 8 and one six sided dice is being used, then the attacker can reach this number by rolling 6-6 as well.
5)For determining the result of the shot, a coin is tossed. If it lands as heads, the player who is shooting scores but if it lands tails, then the other player controls the ball and the game continues.The coin tossing takes place in the same round in which the completing pass was made and in both cases, the other player is in possession of the ball when the game restarts.
For example, let one 6 sided dice be used and the required number of passes be 9. (That is n=6, m=1,p=9) Let us say that A has the ball in the starting.
In the first round, A rolls 4 and B rolls 5. A completes 4 passes within that round.A now has to complete at least 5 more passes in order to get an opportunity to shoot.
In the second round, A rolls 3 and B rolls 3. Possession is overturned and now B controls the ball.B now has 0 passes completed.
In the third round, B rolls 5 and A rolls 2.B has completed 5 passes in this round and needs to complete at least 4 more passes in order to get an opportunity to shoot.
In the fourth round, B rolls 6 and A rolls 4. Although B has overshot the number of passes, due to Rule 4, B is permitted to flip the coin (B however will not be allowed to proceed in this case when question (7) and (8) are being considered. B tosses the coin and gets heads. B scores a goal.B leads by one goal to nil. Now A starts off with possession of the ball.
Now, let p= number of required passes and n=sides of the dice, m= number of dices being used and z=number of goals required to have been scored. Then;
What is the probability that the player in possession scores without being intercepted?
Let both the players use 'm' dice each instead of one. What is the new probability that the player in possession scores without being intercepted? (Here, an interception is only valid if the sum of the numbers obtained by the defender is equal to the sum obtained by the attacker)
What is the minimum number of rounds that need to be completed such that the probability of at least one goal having been scored in the game is maximum? (for any given value of n and p)
(3) but now both the players have 'm' dice.
(3) but with at least 'z' goals having been scored.
(4) but with at least 'z' goals having been scored.
Assume Rule 4 is invalid ie.the player in possession must keep rolling until the desired number is obtained to make the number of passes made exactly equal to the number of required passes. Now find out the answers for (1),(2),(3),(4),(5) and (6) .
Assume Rule 4 is invalid ie. If the player in possession rolls a number that overshoots the required number of passes, then the possession shifts over to the other player. Now find out the answers for (1),(2),(3),(4),(5) and (6) .
Is there any way to solve this question with its generalizations? If yes, how and are there any unique answers? If not, how would one solve for say n=12, m=3 , p=71 and z=5 without plugging in values/ brute force? I understand that is a very long drawn-out question so I'll be grateful to any contributions made.
PS This isn't a homework question/problem so please don't close it because an answer is being requested. It is an original question. Consider this to be a challenge of sorts.
PPS The question has now been segmented. I'll leave this one on as it is the most comprehensive one.
Part I:(The Board Football Problem (Part I))
Part II:(The Board Football Problem (Part II))