You play a TV game in which you have to open one door out of $n$. Behind each door, there is a treasure with probability 1/2, independent of the other doors, so your apriori chance of winning is 1/2.
In order to increase your chances, you can coordinate with a friend. You do the coordination before the game starts. Then, the friend can see what's behind all the doors, and send you a single integer number between 1 and $n$.
You receive your friend's number, but, in addition to that number, you receive $k-1$ more numbers between 1 and $n$, which are selected at random without replacement between 1 and $n$. You don't know which of the $k$ numbers is the real message and which is fake.
What strategy can you and your friend can coordinate, such that your probability of finding a treasure is maximized?
EXAMPLE: suppose $k=2$. Then, one possible strategy is that the friend just selects a door with a treasure behind it, and sends its number to you. You receive your friend's number and another number, in random order. Suppose you just pick the first number that you receive and open the door with that number. If it is your friend's number then you win; if this is a fake number then you have a chance of 1/2 to win. Hence, your total chance of winning with this strategy is 3/4. Is this correct? Is there a better strategy?