Zarball is back!!! This is an easy problem just to revive the Zarball franchise, harder ones will follow ;D
You played Three Games of Zarball and had to win two in a row... you made the right choices (but unfortunately lost).
Your second chance for freedom came with the Five Games of Zarball, in which you had to win at least one of three combinations. Again, you lost in a mistake up against the King.
However you won your freedom, and deserved it, in the intense Royal Zarball Tournament, where you made your way to victory.
Now you have taken the King's offer as the Supreme Dignifiably Appointed Royal Zarball Trainer of Excellence. and figured out the probability of hats being returned to the proper heads in The Royal Zarball Spectators crisis.
You then arranged the Village Zarball Tournament to set up the brackets and byes to make the game as fair as possible in one task, and as skewed toward your buddy Perry as possible in another.
With your diligent success and logical skills, you, as the Supreme Dignifiably Appointed Royal Zarball Trainer of Excellence, have become the King's favorite man on his royal court. He believes your abilities to be unparalleled to any others in the realm. Thus he has assigned you to be in charge of the betting process in the upcoming, much-anticipated match between the realm's Champion and the foreign Champion from the neighboring kingdom. They've never played against each other before and nobody really knows how different the competition is here and abroad, so it's unknown how well evenly the two Champions are matched.
However the King collects a share 1/9 of every successful bet on the realm's Champion, and the foreign Lord collects 2/9 of every successful bet on the foreign Champion (the visiting country is always given the double share). All bets placed are an amount of Crowns (the wealthy spectators cash their gold in for Crowns at the betting booths) on either the realm Champion (RC) or the foreign Champion (FC). If their bet is successful, they are given (from the King's treasury) the amount they bet, but if their bet is a failure, they must pay whatever they bet. Nobody bets more than they have, and all payments are made legally. And of course 1/9 or 2/9 of each successful bet is given to either the King or the foreign Lord.
Three-fifths of the bettors are locals, and two-fifths are foreigners. All bettors have a 3/5 chance to bet on their own Champion and a 2/5 chance to bet on the other country's Champion.
The maximum you can bet is 99 Crowns (they only have two decimal spots down the bet column in the betting record-book), and all of the bettors, eager to display their bet, will always bet the full 99 Crowns.
The King has just realized that he can adjust the tip, weight distribution, tube connectors, etc, of the Zarball Court and effectively decide who wins the game. He comes to you for advice:
(1) You are caught up in the King's greed. Who should win the match, and how many Crowns can the King expect to make/lose out of it, if B is the number of bettors?
(2) You wisely decide to roll a dice to determine the winner, and make it perfectly fair. Ie, if the Foreign Champion winning would bring in 1 Crown per bettor and the Realm Champion winning would bring in 2, the FC would have a 2/3 chance of victory and the RC would have a 1/3. What should the chances be of the RC winning?
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unreality
Zarball is back!!! This is an easy problem just to revive the Zarball franchise, harder ones will follow ;D
You played Three Games of Zarball and had to win two in a row... you made the right choices (but unfortunately lost).
Your second chance for freedom came with the Five Games of Zarball, in which you had to win at least one of three combinations. Again, you lost in a mistake up against the King.
However you won your freedom, and deserved it, in the intense Royal Zarball Tournament, where you made your way to victory.
Now you have taken the King's offer as the Supreme Dignifiably Appointed Royal Zarball Trainer of Excellence. and figured out the probability of hats being returned to the proper heads in The Royal Zarball Spectators crisis.
You then arranged the Village Zarball Tournament to set up the brackets and byes to make the game as fair as possible in one task, and as skewed toward your buddy Perry as possible in another.
With your diligent success and logical skills, you, as the Supreme Dignifiably Appointed Royal Zarball Trainer of Excellence, have become the King's favorite man on his royal court. He believes your abilities to be unparalleled to any others in the realm. Thus he has assigned you to be in charge of the betting process in the upcoming, much-anticipated match between the realm's Champion and the foreign Champion from the neighboring kingdom. They've never played against each other before and nobody really knows how different the competition is here and abroad, so it's unknown how well evenly the two Champions are matched.
However the King collects a share 1/9 of every successful bet on the realm's Champion, and the foreign Lord collects 2/9 of every successful bet on the foreign Champion (the visiting country is always given the double share). All bets placed are an amount of Crowns (the wealthy spectators cash their gold in for Crowns at the betting booths) on either the realm Champion (RC) or the foreign Champion (FC). If their bet is successful, they are given (from the King's treasury) the amount they bet, but if their bet is a failure, they must pay whatever they bet. Nobody bets more than they have, and all payments are made legally. And of course 1/9 or 2/9 of each successful bet is given to either the King or the foreign Lord.
Three-fifths of the bettors are locals, and two-fifths are foreigners. All bettors have a 3/5 chance to bet on their own Champion and a 2/5 chance to bet on the other country's Champion.
The maximum you can bet is 99 Crowns (they only have two decimal spots down the bet column in the betting record-book), and all of the bettors, eager to display their bet, will always bet the full 99 Crowns.
The King has just realized that he can adjust the tip, weight distribution, tube connectors, etc, of the Zarball Court and effectively decide who wins the game. He comes to you for advice:
(1) You are caught up in the King's greed. Who should win the match, and how many Crowns can the King expect to make/lose out of it, if B is the number of bettors?
(2) You wisely decide to roll a dice to determine the winner, and make it perfectly fair. Ie, if the Foreign Champion winning would bring in 1 Crown per bettor and the Realm Champion winning would bring in 2, the FC would have a 2/3 chance of victory and the RC would have a 1/3. What should the chances be of the RC winning?
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