How do you determine the theoretical probability of rolling a five on a die?

Question

Genesis Allen · Accepted Answer

The probability of any event is based on given probabilities of elementary events and a composition of our event as a set of elementary events.  If we know the probabilities of elementary events and a composition of our event, the probability of our event is a sum of probabilities of all elementary events that comprise it. In case of a die, the elementary events are numbers rolled on this die. The so-called "fair" die has all its 6 numbers equally probable. Since the total probability always equals to 1, the probability of each elementary event (rolling each number from 1 to 6) equals to
#P{1}=P{2}=P{3}=P{4}=P{5}=P{6}=1/6#. An event offered in the problem, "rolling a five on a die", contains only one elementary event - number 5, whose probability we know as being equal to #1/6#.  Therefore, the probability of rolling five on a die equals to
#P{5}=1/6#. Just as an example, an event "Rolling a number that is less than five" contains 4 different elementary events - rolling 1, 2, 3 or 4. Each of them has a probability #1/6#. The sum of all elementary events comprising our event, that is the probability of our event, is 
#P{<5}=P{1 or 2 or 3 or 4}=4/6=2/3#.

Ethan Adkins · Answer

undefined The theoretical probability of rolling a five on a fair six-sided die is 1/6.