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Tale Head
Spades Hearts Diamonds Clubs
TC – Mathematics – S2
Coins Die dice
Set/deck of playing cards
PROBABILITIES : intuitive
TC – Mathematics – S2
… ?
Experiment
tossing a coin
Event
it’s a head
Probability
1/2
rolling a die it’s a one 1/6
it’s even 3/6 = 1/2
tossing two coins getting two heads 1/4
rolling two dice it’s a double 6/36 = 1/6
picking 5 playing cards
(from a deck of 32)
i’ve got 3 of a kind 12096/201376
= 54/899 ≈ 0.06
TC – Mathematics – S2
1. Counting
1.1 Reminders on sets
ab
cd
e f
A
ab
ac
dbf
B
A = a ; b ; c ; d ; e ; fB = a ; b ; a ; c ; b ; d ; f
b ∈ A h ∉ Aa ; b ∈ B a ; b ; c ∉ B a ∉ B
Venn diagram
element
subset = part a ; b ; c ⊂ A
TC – Mathematics – S2
1. Counting
1.1 Reminders on sets
a
b
c
Ea
ba
c
a bc
E = a ; b ; c
= ∅ a ; b ∈ P (E)
P (E) = ; a ; b ; c ; a;b ; a;c ; b;c ; a;b;c
P (E)
bc
a bc
a ∉ P (E)
Set of the parts of a set
TC – Mathematics – S2
1. Counting
1.1 Reminders on sets
2 4 6
8 10 12
1 5 3
7 11 9
E
E : integers from 1 to 12
Operations on sets
BA
A : even numbers in E
B : multiples of 3 in E
Complement of A inside E : ; ; ; ; ;A odd 1 3 5 7 9 11= =
Intersection of A and B : ;A B even AND m3 6 12∩ = = ; ; ;A B even AND notm3 2 4 8 10∩ = =Intersection of A and :B
Union of A and B :
; ; ; ; ; ; ;A B even OR m3 2 3 4 6 8 9 10 12∪ = =
; ; ; ; ; ; ;A B odd OR m3 1 3 5 6 7 9 11 12∪ = =Union of and B :A
TC – Mathematics – S2
1. Counting
1.1 Reminders on sets Properties of operations
2 4 6
8 10 12
1 5 3
7 11 9
E
BA
TC – Mathematics – S2
1. Counting
1.1 Reminders on sets Cardinal number of a set
2 4 6
8 10 12
1 5 3
7 11 9
E
BA
Contingency table
TC – Mathematics – S2
1. Counting
1.1 Reminders on sets Cardinal number of a set
Exercise 1
A
B …… = Card(B)
…… = Card( )
………. = Card(A) ……… = Card( ) …… = Card(E)
A
BBA
TC – Mathematics – S2
2.1 Random experiment and events
2. Probabilities
Random experiment
Roll a die,
Note its result
Organize a race,
Note the podium
Sample space
Ω = 1;2;3;4;5;6
A
B
C
D
E
F
G
H
C
FB
Ω = (C,F,B) ; (B,E,A) ;
(E,G,H) ; (D;B;C) ;
… … … … …
Outcomes
Outcomes
TC – Mathematics – S2
2.1 Random experiment and events
2. Probabilities
Events
A : get an even number
Ω = 1;2;3;4;5;6 A = 2;4;6
B : get at least 3
Ω = 1;2;3;4;5;6 B = 3;4;5;6
C : get at maximum 2
Ω = 1;2;3;4;5;6 C = 1;2
Contrary of A : A : everything but A → get an odd number A = 1;3;5
B and C are contrary events : B = C and C = B
A∩B : A AND B : even AND at least 3
Ω = 1;2;3;4;5;6 A∩B = 4;6
A∪B : A OR B : even OR at least 3
Ω = 1;2;3;4;5;6 A∪B = 2;3;4;5;6
B and C are then mutually exclusive
[get 4] and [get less than 3] are mutually exclusive
Ω is certain
∅ is impossible
TC – Mathematics – S2
2.2 Probability on a finite set
2. Probabilities
Event
( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )
; ;p 0 0 p A 1 p 1
p A p A 1
p A B p A B p A
p A B p A p B p A B
∅ = ≤ ≤ Ω =
+ =
∩ + ∩ =
∪ = + − ∩
Owns some outcomes among
the whole set of possibilities
between 0% and 100% of
the possibilitiesA
A is a part of Ω 0 ≤ p(A) ≤ 1
Equalities that are true IN ANY CASE: In case the outcomes are
EQUALLY LIKELY:
( ) ( )( )
Ap A =
ΩCard
Card
EX 17, 18, 19
TC – Mathematics – S2
2.2 Probability on a finite set
2. Probabilities
Conditional probabilities
( ) ( )( )A
p A Bp B
p A
∩=
Given two events A and B not necessarily mutually exclusive
It’s proved that: Special case:
If the information « A occured » doesn’t affect p(B),
so if pA(B) = p(B), then A and B are said INDEPENDENT.
EX 20, 21, 22
Their own probability to occur, without any other information, are p(A) and p(B).
In general : if we do know that A occured, then this information can make
the probability of occurrence of B evoluate. p(B) becomes pA(B).
Equivalent to: p(A ∩ B) = p(A) × p(B)
TC – Mathematics – S2
2.3 Discrete probability distribution
2. Probabilities
8
ΩA
B
C
D
5
0
2
To each event, we set a value.
Each value gets then a
probability to occur.
List of values: X = « variable »
X = « random variable »
« Probability distribution » of X:
xi 8 5 2 0
pi p(A) p(B) p(C) p(D)
Ω is partitioned.
EX 28
H,T
Object
Initial set = issues
Purpose : being able to count the outcomes of an experiment
e.g. : heads or tails
EXPERIMENT Sample space =
outcomes : # ?
Sample space =
HH, HT, TH, TT : # = 4
Toss twice
e.g. : roll a die
1,2,3,4,5,6
Sample space =
111, 112, 113, … : # = ?
Roll 3 times
TC – Mathematics – S2
2. Counting
Reasoning : different cases ?
Order ? Y/N
Repetition ? Y/Nn = # of available elements
p = # of elements to be chosen
Total number of possible outcomes
e.g. : heads or tails
H,T : n = 2
Example of
outcome :
Toss 3 times
p = 3
e.g. : deck of 52 playing cards
… : n = 52
Example of
outcome :
take 3 cards
p = 3
T T H
TC – Mathematics – S2
2. Counting
2.1 p-lists
Order ? Yes
Repetition ? Yesn = # of possibilities at first try = 5
p = # of tries = 2
Outcomes are named « p-lists » (here : 2-lists)
Initial set :
1,2,3,4,5 : n = 5
Outcomes :Form 2-long numbers
p = 2
Number of possible outcomes :
25
TC – Mathematics – S2
2. Counting
Let’s draw a CHOICE TREE :
1st figure1
2
3
4
5
1234512345123451234512345
2nd figure
This 2-leveled tree shows
its 25 ends :
5 times 5 branches = 52
Why 5 ? n = 5
Why 2 levels ? p = 2
Number of p-lists :
52 = np = 25
TC – Mathematics – S2
2. Counting
2.1 p-lists
ORDER
Y N
REPETITIONY p-lists : np
N
Definition : a p-list is an ordered list formed with p elements
taken from a set, with possible repetition.
Result : the number of possible p-lists from a set of n elements is np.
Exercise 7 :
* How many ways for placing 2 objects into 3 drawers ?
* how many numbers of 4 figures only contain the figures 1, 2, 3 ?
* How many words of 5 letters taken from a ; b ; e ; m ; i ; r ; o ?
TC – Mathematics – S2
2. Counting
2.1 p-lists
2.2 Permutations
Order ? Yes
Repetition ? Non = # of possibilities at first try = 5
p = # of tries = 2
Outcomes are named « permutations »
Initial set :
1,2,3,4,5 : n = 5
Outcomes :Form 2-long numbers
With different figures
p = 2
Number of possible outcomes :
20
TC – Mathematics – S2
2. Counting
Let’s draw a CHOICE TREE :
1st figure1
2
3
4
5
23451345124512351234
2nd figureThis 2-leveled tree shows
its 20 ends :
5 times 4 branches = 5×4
Why 5 ? n = 5
Why 4 ? p = 2 ⇒ 2 levels
no repetition
⇒ -1 possibility each next level
number of permutations :
5×4 = 20
TC – Mathematics – S2
2. Counting
2.2 Permutations
ORDER
Y N
REPETITIONY p-lists : np
N Permut.
Definition : a permutation is an ordered list formed with p different
elements taken from a set.
Result : the number of possible permutations of p elements taken
from a set of n elements is
Pnp
P =np n!
(n-p)!
TC – Mathematics – S2
2. Counting
2.2 Permutations
enter n
key : OPTN
screen item : PROB
screen item : nPr
enter p
key : EXE
Casio
enter n
key : MATH
screen item : PRB
screen item : nPr
or Arrangements
enter p
key : ENTER
TI
TC – Mathematics – S2
2. Counting
2.2 Permutations
Exercise 8 :
* how many pairs representative/assistant from a group of 25 students?
* how many ways can 3 blocks be piled, taking them among 10 blocks
of different colors?
* how many words, with 5 different letters in a, b, e, m, i, r, o ?
2.3 Combinations
Order ? No
Repetition ? Non = # of possibilities at first try = 5
p = # of tries = 2
Outcomes are named « combinations »
Initial set :
1,2,3,4,5 : n = 5
Outcomes :Take 2 different figures
p = 2
Number of possible outcomes :
10
TC – Mathematics – S2
2. Counting
A choice tree won’t help us. Let’s compare combinations and permutations :
Combinations : 1,2 ; 1,3 ; 1,4 ; 1,5 ; 2,3 ; 2,4 ; 2,5 ; 3,4 ; 3,5 ; 4,5
# = 10(1,2)
(2,1)
permutations
of the combination
1,2
# = 2
(1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5)
(3,1) (4,1) (5,1) (3,2) (4,2) (5,2) (4,3) (5,3) (5,4)
whole set of permutations
# = 20
Number of combinations :
= 10
TC – Mathematics – S2
2. Counting
2.3 Combinations
ORDER
Y N
REPETITIONY p-lists : np
N Permut. Combin.
Definition : a combination is a set (no order) formed with p different
elements taken from a set.
Result : the number of possible combinations of p elements taken
from a set of n elements is
Pnp
C =np n!
p!(n-p)!
Cpn
TC – Mathematics – S2
2. Counting
2.3 Combinations
enter n
key : OPTN
screen item : PROB
screen item : nCr
enter p
key : EXE
Casio
enter n
key : MATH
screen item : PRB
screen item : nCr
or Combinaisons
enter p
key : ENTER
TI
TC – Mathematics – S2
2. Counting
2.3 Combinations
Exercise 9 :
* How many couples of representatives from a group of 25 students?
* How many different hands of 8 cards from a deck of 32 playing cards ?
* How many drawings of 6 different integers, taking them between 1 and 49 ?
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