Calculate probabilities of success, collision, and Idle I'm doing a research and reached a point that needs knowledge of probabilities. I'm not looking for easy answers, I have watched 300+ minutes tutorial about probability but still not able to answer the questions that I need. 
The following question mimics a part of what I need 
If there are 20 players that are asked to choose a number between 1 to 15:
what is the probability of a number to be chosen by only one player? (success)
what is the number of players that may choose the same number ?(collision)
and what is the probability for numbers that are  not being chosen at all?(Idle)
I appreciate any illustration or recommendation for books or tutorials. 
 A: Let $n$ be one of the numbers $1, 2, \dots, 15$.


*

*The probability that exactly one player will choose $n$:
$$
\underset{\text{which player chooses $n$}}{\underbrace{20}} \times \underset{\text{the probability that said player chooses $n$}}{\underbrace{\frac{1}{15}}}\times\underset{\text{the probability that the other players don't choose $n$}}{\underbrace{\left(\frac{14}{15}\right)^{19}}}
$$

*The probability that at least two players will choose $n$:
$$
1 - \underset{\text{the probability computed in part $3$}}{\underbrace{\left(\frac{14}{15}\right)^{20}}} - \underset{\text{the probability computed in part $1$}}{\underbrace{20 \times \frac{1}{15}\times\left(\frac{14}{15}\right)^{19}}}
$$

*The probability that no player will choose $n$:
$$
\left(\frac{14}{15}\right)^{20}
$$

As per OP's request, here's a more detailed explanation, focusing on the general probability laws underlying the calculations.


*

*Each player is as likely to choose any number as any other number. This means that the probability of player $\#1$ (or any other player) to choose any given number out of the possible fifteen numbers is $\frac{1}{15}$. Therefore, the probability of player $\#1$ (or any other player) to not choose a given number is $1 - \frac{1}{15} = \frac{14}{15}$. It is always the case that, if the probability that some event happens is $p$, then the probability that this event won't happen is $1 - p$.

*The players make their choices independently of each other. This means that in order to calculate the probability that each player makes a certain choice, you multiply the $20$ individual probabilities.

*If you have a list of mutually exclusive alternatives, the probability that one of them will happen is the sum of the individual probabilities. For instance, in part $1$ we consider the $20$ mutually exclusive alternatives:
a) Player $\#1$, and only player $\#1$, will choose the number $n$.
b) Player $\#2$, and only player $\#2$, will choose the number $n$.
...
c) Player $\#20$, and only player $\#20$, will choose the number $n$.
