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}.
As we are commonly going to use set notation, let's review some terms and properties of sets:
We also use the symbol \in to make statements about whether elements are part of the set or not. For example,$2\in\left\{1,2,3\right\}$2∈{1,2,3} and we use the symbol $\notin$∉ to indicate if something is not an element of the set. So $4\notin\left\{1,2,3\right\}$4∉{1,2,3}
the set of numbers on the interval between $1$1 and $5$5 that is$\left\{x\ :\ 1\le x\le5\right\}${x : 1≤x≤5}.
The empty set or the null set, is a set that has no elements in it. There is also a special symbol we use to denote the empty set:$\varnothing$∅.
Sets can refer to coordinates of points. The set $B=\left\{(x,y):\quad y=mx+c\right\}$B={(x,y): y=mx+c}contains the coordinates of points that lie on the line $y=mx+c$y=mx+c.
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 $A\cup B=\left\{4,5,11,12,15,16,17,20,25,30\right\}$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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Consider the sets: $A=\left\{x:\quad x\text{ is an integer}\right\}$A={x: x is an integer} and $B=\left\{-5,2,3,100\right\}$B={−5,2,3,100}. State whether the following statements are true:
a) $B\subset A$B⊂A
b) $B\subseteq A$B⊆A
c) $A\subset B$A⊂B
d) $A\subset A$A⊂A
e) $A\subseteq A$A⊆A
a) This statement is true because all the elements of $B$B are integers, and $A$A contains all the integers, so all the elements in $B$B are also in $A$A.
b) This statement is also true because all the elements of $B$B are integers, and $A$A contains all the integers, so all the elements in $B$B are also in $A$A.
c) This statement is false, because $A$A contains all the integers, which means it contains elements that $B$B does not contain.
d) This statement is false because a proper subset can not equal the original subset.
e) This statement is true because $A$A is a subset of itself, because it is equal to itself which subsets allow.
Determine whether the following points are in the set $A=\left\{(x,y):\quad y=2x-3\right\}.$A={(x,y): y=2x−3}.
a) $(3,4)$(3,4)
b) $(-1,-5)$(−1,−5)
c) $(10,17)$(10,17)
a) To determine whether a point is in the set, we need to see if the point satisfies the equation of the line:
$y$y | $=$= | $2x-3$2x−3 |
$4$4 | $=$= | $2\times3-3$2×3−3 |
$4$4 | $=$= | $2$2 |
Since the left and right hand sides of the equation are not equal, the point does not satisfy the line and is not in the set.
b) To determine whether a point is in the set, we need to see if the point satisfies the equation of the line:
$y$y | $=$= | $2x-3$2x−3 |
$-5$−5 | $=$= | $2\times-1-3$2×−1−3 |
$-5$−5 | $=$= | $-5$−5 |
Since the left and right hand sides of the equation are equal, the point does satisfy the line and is in the set.
a) To determine whether a point is in the set, we need to see if the point satisfies the equation of the line:
$y$y | $=$= | $2x-3$2x−3 |
$17$17 | $=$= | $2\times10-3$2×10−3 |
$17$17 | $=$= | $17$17 |
Since the left and right hand sides of the equation are equal, the point does satisfy the line and is in the set.
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.
If $A$A is the set of factors of $12$12, and $B$B is the set of factors of $18$18, then list the elements of:
$B\cup A$B∪A
$A\cap B$A∩B
The sets $U=\left\{21,8,30,9,28\right\}$U={21,8,30,9,28} and $V=\left\{21,8,30,9,28,7,13,12,26\right\}$V={21,8,30,9,28,7,13,12,26} are such that there are no other elements outside of these two sets.
Is $U$U a subset of $V$V?
Yes
No
State $n(U)$n(U), the order 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 $\left\{21,30,9,28\right\}${21,30,9,28}.
The set $\left\{21,8,30\right\}${21,8,30}.
The empty set $\varnothing$∅.
Consider the diagram below.
List all of the items in:
$A\cap C$A∩C
$\left(B\cap C\right)'$(B∩C)′
$A\cap B\cap C$A∩B∩C