(Solved): Once per minute, a subway train arrives at the XYZ station with probability p. Every time the t ...

Once per minute, a subway train arrives at the $XYZ$ station with probability $p$. Every time the train arrived, it is either Green, Red or Blue. Consecutive minutes are independent: for example, knowing that a train arrived or did not arrive at 10:12AM tells you nothing about what will happen at $10:13\mathrm{AM}$. Trains arriving after 10:00AM always arrive in the following order: Green train always comes first, followed by the Red train, followed by the Blue train. Your train is the Blue Train. You come to the $XYZ$ station at exactly 10:00AM. What is the probability that you will wait exactly 10 minutes for your train? Zara and Sue play the following game. Each of them roll a fair six-sided die once. If Sue's number is greater than or equal to Zara's number, she wins the game. But if Sue rolled a number smaller than Zara's number, then Zara rolls the die again. If Zara's second roll gives a number that is less than or equal to Sue's number, the game ends with a draw. If Zara's second roll gives a number larger than Sue's number, Zara wins the game. Find the probability that Zara wins the game and the probability that Sue wins the game.