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Tale Head

Spades Hearts Diamonds Clubs

TC – Mathematics – S2

Coins Die dice

Set/deck of playing cards

PROBABILITIES : intuitive


… ?

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

Count

PROBABILITIES


Sets


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


1. Counting


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


1. Counting


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


1. Counting

1.1 Reminders on sets Properties of operations


1. Counting

1.1 Reminders on sets Properties of operations

2 4 6

8 10 12

1 5 3

7 11 9

E

BA


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


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


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


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


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


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)


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


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


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


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


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 ?


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


20


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


2. Counting

2.2 Permutations

ORDER

Y N


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)!


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


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


10


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


2. Counting

2.3 Combinations

ORDER

Y N


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


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


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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