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
[beautiful math coming... please be patient]
$\,1\,$, $\,2\,$, $\,3\,$, $\,4\,$, $\,5\,$, 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
[beautiful math coming... please be patient]
$\,1\,$ 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: [beautiful math coming... please be patient] $\,1\,$, $\,2\,$, $\,3\,$, $\,4\,$, $\,5\,$, $\,6\,$ |
$S = \{1,2,3,4,5,6\}$ |
(2) Roll once; record the number that appears on the bottom (hidden) face. |
six possible outcomes: [beautiful math coming... please be patient] $\,1\,$, $\,2\,$, $\,3\,$, $\,4\,$, $\,5\,$, $\,6\,$ |
$S = \{1,2,3,4,5,6\}$ |
(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: [beautiful math coming... please be patient] $\,1\star1\,$, $\,1\star2\,$, $\,1\star3\,$, $\,1\star4\,$, $\,1\star5\,$, $\,1\star6\,$ $\,2\star1\,$, $\,2\star2\,$, $\,2\star3\,$, $\,2\star4\,$, $\,2\star5\,$, $\,2\star6\,$ $\,3\star1\,$, $\,3\star2\,$, $\,3\star3\,$, $\,3\star4\,$, $\,3\star5\,$, $\,3\star6\,$ $\,4\star1\,$, $\,4\star2\,$, $\,4\star3\,$, $\,4\star4\,$, $\,4\star5\,$, $\,4\star6\,$ $\,5\star1\,$, $\,5\star2\,$, $\,5\star3\,$, $\,5\star4\,$, $\,5\star5\,$, $\,5\star6\,$ $\,6\star1\,$, $\,6\star2\,$, $\,6\star3\,$, $\,6\star4\,$, $\,6\star5\,$, $\,6\star6\,$ |
$S = \{$ [beautiful math coming... please be patient] $\,1\star1\,$, $\,1\star2\,$, $\,1\star3\,$, $\,1\star4\,$, $\,1\star5\,$, $\,1\star6\,$, $\,2\star1\,$, $\,2\star2\,$, $\,2\star3\,$, $\,2\star4\,$, $\,2\star5\,$, $\,2\star6\,$, $\,3\star1\,$, $\,3\star2\,$, $\,3\star3\,$, $\,3\star4\,$, $\,3\star5\,$, $\,3\star6\,$, $\,4\star1\,$, $\,4\star2\,$, $\,4\star3\,$, $\,4\star4\,$, $\,4\star5\,$, $\,4\star6\,$, $\,5\star1\,$, $\,5\star2\,$, $\,5\star3\,$, $\,5\star4\,$, $\,5\star5\,$, $\,5\star6\,$, $\,6\star1\,$, $\,6\star2\,$, $\,6\star3\,$, $\,6\star4\,$, $\,6\star5\,$, $\,6\star6\,$ $\}$ |
(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: |
[beautiful math coming... please be patient] $S = \{2,3,4,5,6,7,8,9,10,11,12\}$ |
(5) Roll twice; record the greatest number that appears on the two rolls. |
[beautiful math coming... please be patient] $\,1\,$, $\,2\,$, $\,3\,$, $\,4\,$, $\,5\,$, $\,6\,$ 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 | [beautiful math coming... please be patient] $\displaystyle\frac{n(E)}{n(S)} = \frac16$ | $P(x = 3) = \frac16$ read as: “The probability that $\,x\,$ is $\,3\,$ is $\,\frac16\,$.” |
$E = \{4\}$ | getting a $\,4\,$ on a single roll of a fair die | [beautiful math coming... please be patient] $\displaystyle\frac{n(E)}{n(S)} = \frac16$ | $P(x = 4) = \frac16$ |
$E = \{3,4\}$ | 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 | [beautiful math coming... please be patient] $\displaystyle\frac{n(E)}{n(S)} = \frac36 = \frac12$ | $P(x \text{ is even}) = \frac36 = \frac 12$ |
$E = \{2,3,4,5,6\}$ | getting a number greater than $\,1\,$on a single roll of a fair die | [beautiful math coming... please be patient] $\displaystyle\frac{n(E)}{n(S)} = \frac56$ | $P(x \gt 1) = \frac56$ |