Maths Statistics
Give answers to 4sf
Probability
A and B are independent: if I know A will/won’t happen then this won’t affect my estimate
of P(B) and vice versa
P ( A ∪B ) =P ( A )+ P ( B )−P ( A ∩ B )
Mutually exclusive events cannot occur simultaneously
Complementary events sum to 1
A and B are mutually exclusive then P ( A ∩B )=0
A and B are complementary events then P ( A ) + P ( B ) = 1
Notation:
o ∈means is an element of
o ∉ means not an element of
o ∪means union
o ∩ means intersection
o X ∨X ' means the complement of the set X
o n( X ) means the number of elements in set X
o A ⊂B means A is a proper subset of B (cannot be same set)
o A ⊆B means A is a subset of B
o ∅ means a null set
o A' ∪B' ∪C' =( A ∩B ∩C )'
Venn diagrams or contingency tables (two-way tables) can be used to display events
Contingency table:
Central tendency measures averages
Spread measures variability
Boxplots:
,
P ( A ∩ B)
P ( A|B )=
P (B )
If 2 events are independent then
o P ( R ∩ S )=P ( R ) P ( S )
o This can be derived from conditional probability:
P ( A ∩ B)
P ( A|B )=
P (B )
If independent then P ( A|B ) =P ( A )
Substituting this in:
P(A∩B)
P ( A )=
P ( B)
P ( A ∩B )=P ( A ) P(B)
Draw tree diagrams!
There are n! ways to arrange n items
n!
If there are r spaces and n items altogether, there are ways to arrange them, IF
( n−r ) !
order matters (permutation- nPr)
n!
If there are x spaces and n items altogether, there are ways to arrange them,
( n−r ) ! r !
n
removing repeats IF order does not matter (combination nCr or ( ) ¿
r
If there are repeated items, then you divide the total by the factorial of the number of
repeats:
o Eg. Finding anagrams of the word LOLLIPOP
o There are 8 letters
o There are 2 repeated Os, 3 repeated Ls, 2 repeated Ps
, 8!
o Therefore, there are anagrams
2!× 3! × 2!
20
C3 ×2 is not the same as 40C6
Sigma
max
∑ counter=start + ( start +1 ) + ( start +2 ) +…+ ( max−1 ) +max
counter=start
4
eg . ∑ x 2=22 +32 + 42=29
x=2
n n
∑
❑
❑is the same as ∑ i
i=1
n n
∑
❑
k=∑ k=¿ k+ k +k + …+k =nk ¿
i=1
∑ k x i=k ∑ xi
∑ ( xi + y i ) =∑ x i+∑ y i
, Binomial Distribution
k n−k
P ( E )=nCk × p × q where
o n is the total number of trials
o k is the number of trials satisfying the event
o p is the probability of the event occurring
o q is the probability of the event not occurring
n!
o Remember, nCk=
r ! ( n−r ) !
To apply binomial distribution the following conditions must be met:
o There are a fixed number of trials (n)
o Every trial has only two possible results, success or failure
o The probability of success for each trial is constant (p)
o The trials are independent
You can define a random variable for binomial probability
o Eg. X is the number of 5s rolled after 5 rolls of a die
1
X Bin(5 , )
6
To then indicate the probability of a specific number of the event occurrence, you write:
2 3
1 5
Eg. P ( X=2 )=5 C 2× ×
6 6
Binomial distribution is discrete so it must be drawn as a bar chart, not a smooth curve
E ( X ) =np
Var(X) = npq=np (1− p)
Give answers to 4sf
Probability
A and B are independent: if I know A will/won’t happen then this won’t affect my estimate
of P(B) and vice versa
P ( A ∪B ) =P ( A )+ P ( B )−P ( A ∩ B )
Mutually exclusive events cannot occur simultaneously
Complementary events sum to 1
A and B are mutually exclusive then P ( A ∩B )=0
A and B are complementary events then P ( A ) + P ( B ) = 1
Notation:
o ∈means is an element of
o ∉ means not an element of
o ∪means union
o ∩ means intersection
o X ∨X ' means the complement of the set X
o n( X ) means the number of elements in set X
o A ⊂B means A is a proper subset of B (cannot be same set)
o A ⊆B means A is a subset of B
o ∅ means a null set
o A' ∪B' ∪C' =( A ∩B ∩C )'
Venn diagrams or contingency tables (two-way tables) can be used to display events
Contingency table:
Central tendency measures averages
Spread measures variability
Boxplots:
,
P ( A ∩ B)
P ( A|B )=
P (B )
If 2 events are independent then
o P ( R ∩ S )=P ( R ) P ( S )
o This can be derived from conditional probability:
P ( A ∩ B)
P ( A|B )=
P (B )
If independent then P ( A|B ) =P ( A )
Substituting this in:
P(A∩B)
P ( A )=
P ( B)
P ( A ∩B )=P ( A ) P(B)
Draw tree diagrams!
There are n! ways to arrange n items
n!
If there are r spaces and n items altogether, there are ways to arrange them, IF
( n−r ) !
order matters (permutation- nPr)
n!
If there are x spaces and n items altogether, there are ways to arrange them,
( n−r ) ! r !
n
removing repeats IF order does not matter (combination nCr or ( ) ¿
r
If there are repeated items, then you divide the total by the factorial of the number of
repeats:
o Eg. Finding anagrams of the word LOLLIPOP
o There are 8 letters
o There are 2 repeated Os, 3 repeated Ls, 2 repeated Ps
, 8!
o Therefore, there are anagrams
2!× 3! × 2!
20
C3 ×2 is not the same as 40C6
Sigma
max
∑ counter=start + ( start +1 ) + ( start +2 ) +…+ ( max−1 ) +max
counter=start
4
eg . ∑ x 2=22 +32 + 42=29
x=2
n n
∑
❑
❑is the same as ∑ i
i=1
n n
∑
❑
k=∑ k=¿ k+ k +k + …+k =nk ¿
i=1
∑ k x i=k ∑ xi
∑ ( xi + y i ) =∑ x i+∑ y i
, Binomial Distribution
k n−k
P ( E )=nCk × p × q where
o n is the total number of trials
o k is the number of trials satisfying the event
o p is the probability of the event occurring
o q is the probability of the event not occurring
n!
o Remember, nCk=
r ! ( n−r ) !
To apply binomial distribution the following conditions must be met:
o There are a fixed number of trials (n)
o Every trial has only two possible results, success or failure
o The probability of success for each trial is constant (p)
o The trials are independent
You can define a random variable for binomial probability
o Eg. X is the number of 5s rolled after 5 rolls of a die
1
X Bin(5 , )
6
To then indicate the probability of a specific number of the event occurrence, you write:
2 3
1 5
Eg. P ( X=2 )=5 C 2× ×
6 6
Binomial distribution is discrete so it must be drawn as a bar chart, not a smooth curve
E ( X ) =np
Var(X) = npq=np (1− p)
