How do I summarize the possible outcomes of a random variable and their associated probabilities?

Your choice of random experiment will determine how you proceed.

Explanation: 
Tossing two coins is a random experiment. It outcome are - 
Head on first coin Head on second coin - HH 
Head on first coin Tail on second coin - HT 
Tail on first coin Head on second coin - TH 
Tail on first coin Tail on second coin - TT 
{HH, HT, TH, TT} 
It can be utilized in a real-world scenario. Let's say two salespeople are dispatched to make sales. [This again is a random experiment. Its equivalent is tossing two coins.] Their outcome can be summarized as - 
1st Sales Man Success 2nd Sales Man Success - SS 
1st Sales Man Success 2nd Sales Man Fail - SF 
1st Sales Man Fail 2nd Sales Man Success - FS 
1st Sales Man Fail 2nd Sales Man Fail - FF 
{ SS, SF, FS, FF} 
This is how You can summarize the outcomes. 
First try with simple experiment then move to advanced experiments.