There are $n$ players who play a game of selecting numbers from a range $k$ over a period of time and the rules of the game are as follows :
- The game is played for 1 minute and the players are only allowed to select a random number from the range $k$ after the passage of every 1 second.
- After selecting a number, a player is not allowed to select another number until the selected number decreases to 0. It means that after selecting a number, the number decreases by $1$ after every second. As an example, if a player $\mathcal A$ selects a the number $5$, then $\mathcal A$ has to wait for $5$ seconds before the number becomes $0$ and only then $\mathcal A$ can select another number.
I want to find following probabilities :
Case 1: When a particular player $\mathcal A$ selects a number from a range $k$, what is the probability that other players also selects the same number from the given range $k$
Case 2: Consider a player $\mathcal A$, who selects 5. Then after 1 second, what is the probability that other players select a number which is equal to the (selected number by $\mathcal A$ (minus $1$) i.e. $5 - 1 = 4$). Similarly after $2$ seconds, what is the probability that any other player selects a number which is equal to the (selected number by $\mathcal A$ (minus $2$) i.e. $5 - 2 = 3$). And so on until the number selected by $\mathcal A$ becomes $0$.
In short I want to find the combined probability of selecting a number, which is similar to the number selected by $\mathcal A$ (both in Case 1, and while the number is being decremented in Case 2), by other players for 1 minute.
I am bad at probabilities , so a stepwise approach would be very helpful.