Basic things you need to know about sets and probability

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Basic things you need to know about sets and probability
Symbol
or { }
U
'
∪
⋂

Name
Typical Example
Trickier Example
empty set
{ } ={humans on Mars} (True)
= { } (False)
is an element of
2
{1, 2, 3} (True)
not an element of
2
{1, 2, 3} (False)
is a subset of
{a, b} {a, b, c} (True)
{a, b, c} {a, b, c} (True)
a proper subset of
{a, b}
{a, b, c} (True)
{a, b, c} {a, b, c} (False)
universal set
complement
If U={1, 2, 3} & A={1}, A'={2,3}
union
If A={p,q}, B={r,s}, A∪B={p,q,r,s}
intersection
If A={p,q,r}, B={q,r,s}, A⋂B={q,r}
Two sets are equal if and only if they have the same elements (and order does not
Other comments
{} is a subset of every set
every set is a subset of itself
no set is a proper subset of itself
the set of all elements possible
must know U to find complement
matter).
Ex: {1, 3, 5, 7} = {3, 1, 7, 5} but {1, 3, 5} ≠ {1, 3, 7}

A set with n elements has

Union Rule for Sets:

Union Rule for Probability: P(A∪B) = P(A) + P(B) – P(A⋂B)

Complement Rule: P(E) = 1 – P(E') and P(E') = 1 – P(E)

Basic Probability Principle: If E is an event and S is its equally likely sample space, then
subsets.
n(A∪B) = n(A) + n(B) – n(A⋂B)
( )
( )
( )
( )

Odds:

A practical way of calculating odds and how odds relate to probability:
(
)
Think of the sample space as being divided into two parts…
o
The things that you want to happen…call this “The part you want” and
o
The things that you don’t want to happen (everything else)”…call this “The part you don’t want”
o
So, n(part you want) + n(part you don’t want”) = n(WHOLE sample space)
(You don’t literally have to want these events to happen…it’s just a nice way of phrasing it in my opinion.)
o
From this point of view, odds are always calculated using the “parts”:
o
To find the odds against, just flip the numbers in the odds in favor calculation
o
While odds are always “part to part”, probability is always “part to whole”:
( )
(
(
)
)
(
(
)
(
)
)
(
(
)
)

You need to know how to write out the sample space for a problem involving rolling 2 dice:
11
21
31
41
51
61

Conditional Probability:
o
12
22
32
42
52
62
(
)
13
23
33
43
53
63
(
14
24
34
44
54
64
15
25
35
45
55
65
16
26
36
46
56
66
)
( )
A practical way of calculating conditional probabilities:
Think of the “condition” as something that REDUCES the original sample space and once the REDUCED sample
space is determined, the conditional probability can be calculated as usual by simply using the Basic
Probability Principle and the REDUCED sample space in place of the original sample space.

Product Rule of Probability: P(A⋂B) = P(A) · P(B A)
[ or, if you prefer: P(A⋂B) = P(B) · P(A
Note: If A and B are independent events, then this simply becomes : P(A⋂B) = P(A) · P( B)

The Product Rule for Probability can perhaps best be visualized via a tree diagram:
Tree
diagram
illustrating
Product Rule
of
Probability
for
Dependent
Events.
Tree
diagram
illustrating
Product Rule
of
Probability
for
Independent
Events.
P(A⋂B)
= P(A)·P(B A)
(multiplying
down the
appropriate
branch of the
tree)
P(A⋂B)
= P(A)·P(B)
(multiplying
down the
appropriate
branch of the
tree)
B) ]
Sets and Venn Diagrams
A
A⋂B
A∪B
A⋂B⋂C
A⋂B' [or A-B]
A'
(A⋂B)' [or A'∪B']
(A∪B)' [or A'⋂B']
A'⋂B'⋂C' [or (A∪B ∪C)']
B⋂A' [or B-A]
You may have noticed that when you complement a union or intersection, each set gets complemented and the
operation switches (union becomes intersection and intersection becomes union).
For example,
(A∪B)' = A'⋂B'
(A∪B ∪C)' = A'⋂B'⋂C'
(A⋂B)' = A'∪B'
(A∪B ∪C)' = A'⋂B'⋂C'
So, if you need a Venn representation of one of these, use the one that seems easiest.
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