Probability is the area of mathematics devoted to predicting the likelihood of uncertain occurrences.
For example, when you roll a die, it is uncertain whether you'll see a
$\,1\,$, $\,2\,$, $\,3\,$, $\,4\,$, $\,5\,$, or $\,6\,$.
However, it is possible to talk about how likely it is for each number to occur.
A die is a cube with the numbers
$\,1\,$ through $\,6\,$ represented on its six faces.
When it is thrown (‘rolled’), one of these six faces appears on top. For a fair die, each of the numbers $\,1\,$ through $\,6\,$ is equally likely to occur. 
There are lots of experiments involving a single fair die.
Here are some of them, with their corresponding outcomes and sample spaces:
EXPERIMENT  OUTCOMES  SAMPLE SPACE 
(1)
Roll once; record the number that appears on the top face. 
six possible outcomes: $\,1\,$, $\,2\,$, $\,3\,$, $\,4\,$, $\,5\,$, $\,6\,$ 
$S = \{1,2,3,4,5,6\}$ 
(2)
Roll once; record the number that appears on the bottom (hidden) face. 
six possible outcomes: $\,1\,$, $\,2\,$, $\,3\,$, $\,4\,$, $\,5\,$, $\,6\,$ 
$S = \{1,2,3,4,5,6\}$ 
(3)
Roll twice; record the numbers on each of these rolls, in order. In this experiment and those below, use the notation $\,1\star5\,$ to denote a roll of $\,1\,$, followed by a roll of $\,5\,$. 
$36\,$ possible outcomes: $\,1\star1\,$, $\,1\star2\,$, $\,1\star3\,$, $\,1\star4\,$, $\,1\star5\,$, $\,1\star6\,$ $\,2\star1\,$, $\,2\star2\,$, $\,2\star3\,$, $\,2\star4\,$, $\,2\star5\,$, $\,2\star6\,$ $\,3\star1\,$, $\,3\star2\,$, $\,3\star3\,$, $\,3\star4\,$, $\,3\star5\,$, $\,3\star6\,$ $\,4\star1\,$, $\,4\star2\,$, $\,4\star3\,$, $\,4\star4\,$, $\,4\star5\,$, $\,4\star6\,$ $\,5\star1\,$, $\,5\star2\,$, $\,5\star3\,$, $\,5\star4\,$, $\,5\star5\,$, $\,5\star6\,$ $\,6\star1\,$, $\,6\star2\,$, $\,6\star3\,$, $\,6\star4\,$, $\,6\star5\,$, $\,6\star6\,$ 
$S = \{$ $\,1\star1\,$, $\,1\star2\,$, $\,1\star3\,$, $\,1\star4\,$, $\,1\star5\,$, $\,1\star6\,$, $\,2\star1\,$, $\,2\star2\,$, $\,2\star3\,$, $\,2\star4\,$, $\,2\star5\,$, $\,2\star6\,$, $\,3\star1\,$, $\,3\star2\,$, $\,3\star3\,$, $\,3\star4\,$, $\,3\star5\,$, $\,3\star6\,$, $\,4\star1\,$, $\,4\star2\,$, $\,4\star3\,$, $\,4\star4\,$, $\,4\star5\,$, $\,4\star6\,$, $\,5\star1\,$, $\,5\star2\,$, $\,5\star3\,$, $\,5\star4\,$, $\,5\star5\,$, $\,5\star6\,$, $\,6\star1\,$, $\,6\star2\,$, $\,6\star3\,$, $\,6\star4\,$, $\,6\star5\,$, $\,6\star6\,$ $\}$ 
(4)
Roll twice; record the sum of the numbers that appear. 
this results only from rolling a $\,1\,$ followed by a $\,1\,$. The largest sum you can get is $\,12\,$; this results only from rolling a $\,6\,$ followed by a $\,6\,$. Convince yourself that every whole number between $\,2\,$ and $\,12\,$ is also a possible outcome; for example, you could get $\,5\,$ in all these ways: 
$S = \{2,3,4,5,6,7,8,9,10,11,12\}$ 
(5)
Roll twice; record the greatest number that appears on the two rolls. 
$\,1\,$, $\,2\,$, $\,3\,$, $\,4\,$, $\,5\,$, $\,6\,$ For $\,1\star1\,$, the greatest number is $\,1\,$. This is the only way to get an outcome of $\,1\,$. For $\,1\star2\,$, the greatest number is $\,2\,$. There are three ways to get the outcome $\,2\,$: For $\,1\star3\,$, the greatest number is $\,3\,$. There are five ways to get the outcome $\,3\,$: 
$S = \{1,2,3,4,5,6\}$ 
Some sample spaces are much easier to work with than others.
In experiments (1), (2) and (3) above, each member of the sample space is equally likely:
When a sample space has equally likely outcomes, then computing probabilities is as easy as counting:
This is best illustrated by an example.
Let's consider the first (or second) experiment above—a single roll of a fair die.
The example below also clarifies the idea of an ‘event’,
and illustrates notation that is frequently used in connection with
probability problems.
event  interpretation of event  probability of event  some conventional language used to report the probability 
$E = \{3\}$  getting a $\,3\,$ on a single roll of a fair die  $\displaystyle \cssId{s118}{\frac{n(E)}{n(S)}} \cssId{s119}{= \frac16}$ 
$\cssId{s120}{P(x = 3)}
\cssId{s121}{= \frac16}$ read as: “The probability that $\,x\,$ is $\,3\,$ is $\,\frac16\,$.” 
$E = \{4\}$  getting a $\,4\,$ on a single roll of a fair die  $\displaystyle\frac{n(E)}{n(S)} = \frac16$  $P(x = 4) = \frac16$ 
$E = \{3,4\}$  getting a $\,3\,$ or a $\,4\,$ on a single roll of a fair die  $\displaystyle\frac{n(E)}{n(S)} = \frac26 = \frac13$  $P(x = 3 \text{ or } x=4) = \frac26 = \frac 13$ 
$E = \{2,4,6\}$  getting an even number on a single roll of a fair die  $\displaystyle\frac{n(E)}{n(S)} = \frac36 = \frac12$  $P(x \text{ is even}) = \frac36 = \frac 12$ 
$E = \{2,3,4,5,6\}$  getting a number greater than $\,1\,$on a single roll of a fair die  $\displaystyle\frac{n(E)}{n(S)} = \frac56$  $P(x \gt 1) = \frac56$ 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
