Before we calculate probabilities, let's familiarise ourselves with the language and notation for describing components in this topic. In probability, the sample space is a list of all the possible outcomes of an experiment.
Outcomes are the results of an experiment or trial. For example, think about flipping a coin. There are two possible outcomes - a head or a tail. So when we list (or write out) the sample space, we write:
$S=\left\{heads,tails\right\}$S={heads,tails}
We can write a sample space using a list, table, set notation as above or a diagram such as a Venn or Tree diagram.
An event is a subset of the of the sample space and is often represented by a capital letter to abbreviate the description of the event in calculations. For example we could have the experiment of rolling a six-sided dice and let $E=\text{the event of rolling an even number}$E=the event of rolling an even number, $O=\text{the event of rolling an odd number}$O=the event of rolling an odd number and $A=\text{the event of rolling a number less than 3}$A=the event of rolling a number less than 3.
Then our sample space is $S=\left\{1,2,3,4,5,6\right\}$S={1,2,3,4,5,6} and the events can be written as the sets $E=\left\{2,4,6\right\}$E={2,4,6}, $O=\left\{1,3,5\right\}$O={1,3,5} and $A=\left\{1,2\right\}$A={1,2}.
The notations commonly used in probability are related to function notation and also to the notations of set theory. This is because the range of possible outcomes of an experiment or observation is a set. We assign numbers - called probabilities - to subsets of these outcomes in the manner of a function.
Thus, the notation $P\left(A\right)$P(A) means 'The probability that event $A$A occurs'.
As we are commonly going to use set notation, let's review some terms and properties of sets:
Just like how a road intersection is the place where two roads cross paths, an intersection of sets is where two sets overlap. Elements that appear in the intersection of sets are elements that have the same characteristic as both the individual sets.
Mathematically we write the intersection of sets using the intersection symbol, $\cap$∩. We interpret the intersection of $A$A and $B$B, $A\cap B$A∩B to be what appears in both set $A$A and set $B$B. It helps some students to relate $\cap$∩ to AND or to think of the symbol like a bridge joining both sets.
$A=\left\{5,10,15,20,25,30\right\}$A={5,10,15,20,25,30} and $B=\left\{6,12,18,24,30\right\}$B={6,12,18,24,30} then $A\cap B=\left\{30\right\}$A∩B={30}
If we consider the intersection the 'and' of mathematical sets, then the union is the OR. $A\cup B$A∪B is the notation we use, and we would read this as the union of $A$A and $B$B. It is the set of the elements that are in either $A$A or $B$B.
$A=\left\{5,11,16,17,20,25\right\}$A={5,11,16,17,20,25} and $B=\left\{4,12,15,25,30\right\}$B={4,12,15,25,30} then $A\cup B=4,5,11,12,15,16,17,20,25,30$A∪B=4,5,11,12,15,16,17,20,25,30
A complement of an event are all outcomes that are NOT the event. If $A$A is the event then the complement is denoted $A'$A′ or sometimes $\overline{A}$A.
The following are examples of events and their complements:
If events are mutually exclusive, it means they cannot happen at the same time.
Some examples of experiments that involve mutually exclusive events are:
Since these events cannot both occur at the same time, they are mutually exclusive events.
However, some events can happen at the same time and we call this non-mutually exclusive. For example:
Since these events can both occur at the same time, they are not mutually exclusive events.
A Venn Diagram is a pictorial way to display relationships between different sets. The idea of a Venn diagram was first introduced by John Venn in the late 1800's and they are still one of the most powerful visualisations for relationships.
For the numbers between $2$2 and $20$20, let $E=\left\{\text{even numbers}\right\}$E={even numbers}, and $M=\left\{\text{multiplies of 3}\right\}$M={multiplies of 3}.
Place the numbers in the appropriate sections, for each ask .... Is the number even? Is it a multiple of $3$3? Is it both? or Is it none of those options?
Take note of how the numbers that do not fit into either set are placed outside the circles, but still within the bounds of the universal set.
Using Venn Diagrams when solving problems about sets will need us to be able to identify using set notation the regions in the Venn Diagram. The following applet will let you explore the different regions.
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A standard six-sided die is rolled.
List the sample space.
(Separate outcomes with a comma)
List the sample space for rolling a number strictly less than $3$3. Separate outcomes with a comma.
List the sample space for rolling a number divisible by $3$3. Separate outcomes with a comma.
List the sample space for rolling an even number. Separate outcomes with a comma.
Two events are defined as:
Event $A$A: it will rain tomorrow
Event $B$B: there will be a storm tomorrow
The notation $P(A\cup B)$P(A∪B) is suitable to describe which of the following probabilities?
probability of no rain tomorrow
probability of there being either a storm or rain tomorrow
probability of a storm occurring tomorrow
probability of there being a storm tomorrow but no rain
How could the probability of the event $A$A: "getting a number greater than 3 when a die is rolled" be written using probability notation?
Select the three the correct options.
$P\left(A\right)$P(A)
$P$P$($(rolling a die$)$)
$50$50
$1-P\left(A'\right)$1−P(A′)
$P$P$($(greater than 3$)$)
$E\left(A\right)$E(A)
Consider the two events:
A: Paul wins the golf tournament
B: Paul wins the badminton tournament
The probability that Paul wins either the golf or badminton but not both can be represented by:
$P\left(A'\cap B'\right)$P(A′∩B′)
$P\left(\left(A\cap B'\right)\cup\left(B\cap A'\right)\right)$P((A∩B′)∪(B∩A′))
$P\left(A\cup B\right)$P(A∪B)
$P\left(A'\cap B\right)$P(A′∩B)
The sets $U=\left\{20,8,26,3,15\right\}$U={20,8,26,3,15} and $V=\left\{20,8,26,3,15,2,24,10,27\right\}$V={20,8,26,3,15,2,24,10,27} are such that there are no other elements outside of these two sets.
Is $U$U a subset or proper subset of $V$V?
A subset.
A proper subset.
State the cardinality of $U$U.
List the elements of $U'$U′.
List the elements of the universal set. State the elements on the same line, separated by a comma.
Which set is $V'$V′?
The set $\left\{20,8,26\right\}${20,8,26}.
The empty set $\varnothing$∅.
The set $\left\{20,26,3,15\right\}${20,26,3,15}.