Set/deck of playing cards. Spades Hearts Diamonds Clubs

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1 TC Mathematics S2 Coins Die dice Tale Head Set/deck of playing cards Spades Hearts Diamonds Clubs

2 TC Mathematics S2 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/ = 54/

3 TC Mathematics S2 Sets Count PROBABILITIES

4 TC Mathematics S2 1. Counting 1.1 Reminders on sets A B Venn diagram a c b e d f a b d b f a c element subset = part A = {a ; b ; c ; d ; e ; f} b A h A {a ; b ; c} A B = {{a ; b} ; {a ; c} ; {b ; d ; f}} {a ; b} B {a ; b ; c} B a B

5 TC Mathematics S2 1. Counting 1.1 Reminders on sets Set of the parts of a set b a c E = {a ; b ; c} E a b a b a c c b c a b c P (E) P (E) = {{} ; {a} ; {b} ; {c} ; {a;b} ; {a;c} ; {b;c} ; {a;b;c}} {} = {a ; b} P (E) a P (E)

6 TC Mathematics S2 1. Counting 1.1 Reminders on sets A E : integers from 1 to 12 A : even numbers in E B : multiples of 3 in E E B Operations on sets 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 Union of A and B : { } { } { } { ; ; ; ; ; ; ; } A B = odd OR m3 =

7 TC Mathematics S2 1. Counting 1.1 Reminders on sets Properties of operations

8 TC Mathematics S2 1. Counting 1.1 Reminders on sets Properties of operations A E B

9 TC Mathematics S2 1. Counting 1.1 Reminders on sets Cardinal number of a set A E B Contingency table

10 TC Mathematics S2 1. Counting 1.1 Reminders on sets Cardinal number of a set Exercise 1 B B A A = Card(B) = Card( B). = Card(A) = Card( A) = Card(E)

race, Note the podium A B C D E F G H B C F

11 TC Mathematics S2 2. Probabilities 2.1 Random experiment and events Random experiment Sample space Roll a die, Note its result Ω = {1;2;3;4;5;6} Outcomes Organize a race, Note the podium A B C D E F G H B C F Ω = { (C,F,B) ; (B,E,A) ; (E,G,H) ; (D;B;C) ; } Outcomes

12 TC Mathematics S2 2. Probabilities 2.1 Random experiment and 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} Events 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} C : get at maximum 2 Ω = {1;2;3;4;5;6} C = {1;2} Ω is certain is impossible 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 B and C are then mutually exclusive [get 4] and [get less than 3] are mutually exclusive

13 TC Mathematics S2 2. Probabilities 2.2 Probability on a finite set EX 17, 18, 19 Event A Owns some outcomes among the whole set of possibilities between 0% and 100% of the possibilities A is a part of Ω 0 p(a) 1 Equalities that are true IN ANY CASE: ( ) ; ( ) ; ( ) p = 0 0 p A 1 p Ω = 1 ( ) + ( ) = ( ) + ( ) = ( ) p A p A 1 p A B p A B p A ( ) = ( ) + ( ) ( ) p A B p A p B p A B In case the outcomes are EQUALLY LIKELY: ( ) p A = Card Card ( A) ( Ω)

14 TC Mathematics S2 2. Probabilities 2.2 Probability on a finite set Conditional probabilities EX 20, 21, 22 Given two events A and B not necessarily mutually exclusive 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 p A (B). It s proved that: A ( ) p B = ( ) p( A) p A B Special case: If the information «A occured» doesn t affect p(b), so if p A (B) = p(b), then A and B are said INDEPENDENT. Equivalent to: p(a B) = p(a) p(b)

15 TC Mathematics S Probabilities 2.3 Discrete probability distribution 8 5 A D 2 C Ω B EX 28 Ω is partitioned. To eachevent, weset a value. List of values: X = «variable» Eachvalue getsthena probability to occur. X = «random variable» «Probability distribution» of X: x i p i p(a) p(b) p(c) p(d)

Initial set = {issues} EXPERIMENT Sample space = {outcomes} : #? e.g.

16 TC Mathematics S2 2. Counting Purpose: beingable to count the outcomesof an experiment Object Initial set = {issues} EXPERIMENT Sample space = {outcomes} : #? e.g. : headsor tails e.g. : roll a die {H,T} {1,2,3,4,5,6} Toss twice Roll 3 times Sample space = {HH, HT, TH, TT} :# = 4 Sample space = {111, 112, 113, }:# =?

17 TC Mathematics S2 Reasoning: different cases? e.g. : headsor tails 2. Counting {H,T} : n = 2 e.g. : deckof 52 playingcards { } : n = 52 Toss3 times p = 3 take3 cards p = 3 n= # of availableelements p= # of elementsto bechosen Example of outcome: Example of outcome: T T H Repetition? Y/N Order? Y/N Total number of possible outcomes

18 TC Mathematics S2 2.1 p-lists 2. Counting Initial set : {1,2,3,4,5} : n = 5 Form 2-long numbers p = 2 Outcomes: Number of possible outcomes: 25 n= # of possibilitiesatfirst try= 5 p= # of tries = 2 Repetition? Yes Order? Yes Outcomesare named«p-lists» (here: 2-lists)

19 TC Mathematics S2 2.1 p-lists 1st figure 2. Counting Let sdrawa CHOICE TREE : 2nd figure This 2-leveled tree shows its25 ends : 5 times 5 branches = 5 2 Why5? n = 5 Why2 levels? p = 2 Number of p-lists: 5 2 = n p = 25

20 TC Mathematics S2 2.1 p-lists 2. Counting REPETITION ORDER Y Y p-lists: np N N Definition: a p-list is an ordered list formed with p elements taken from a set, with possible repetition. Result: the numberof possible p-listsfroma set of nelementsisn p. Exercise7 : * How manywaysfor placing2 objectsinto3 drawers? * how many numbers of 4 figures only contain the figures 1, 2, 3? * How manywordsof 5 letterstakenfrom{a ; b ; e ; m ; i ; r ; o}?

21 TC Mathematics S2 2. Counting 2.2 Permutations Initial set : {1,2,3,4,5} : n = 5 Form 2-long numbers With different figures p = 2 Outcomes: Number of possible outcomes: 20 n= # of possibilitiesatfirst try= 5 p= # of tries = 2 Repetition? No Order? Yes Outcomes are named«permutations»

22 TC Mathematics S2 2. Counting 2.2 Permutations Let sdrawa CHOICE TREE : 1st figure nd figure This 2-leveled tree shows its20 ends : 5 times 4 branches = 5 4 Why5? n = 5 Why4? p = 2 2 levels no repetition -1 possibility each next level number of permutations : 5 4 = 20

23 TC Mathematics S2 2. Counting 2.2 Permutations Y ORDER N REPETITION Y p-lists: np N Permut. P n p 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 froma set of nelementsis p n! (n-p)! P = n

24 TC Mathematics S2 Casio 2. Counting 2.2 Permutations Exercise8 : * 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}? enter n key : OPTN screenitem : PROB screenitem : npr enter p key : EXE TI enter n key : MATH screenitem : PRB screenitem : npr or Arrangements enter p key : ENTER

25 TC Mathematics S2 2. Counting 2.3 Combinations Initial set : {1,2,3,4,5} : n = 5 Take 2 different figures p = 2 Outcomes: Number of possible outcomes: 10 n= # of possibilitiesatfirst try= 5 p= # of tries = 2 Repetition? No Order? No Outcomes are named«combinations»

26 TC Mathematics S2 2. Counting 2.3 Combinations 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) (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (2,1) (3,1) (4,1) (5,1) (3,2) (4,2) (5,2) (4,3) (5,3) (5,4) permutations of the combination {1,2} # = 2 whole set of permutations # = 20 Number of combinations: = 10

27 TC Mathematics S2 2. Counting 2.3 Combinations Y ORDER N REPETITION Y p-lists: np N Permut. P n p Combin. C p n 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 froma set of nelementsis p n! p!(n-p)! C = n

TC Mathematics S2 2. Counting 2.3 Combinations Exercise9 : * 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?

28 TC Mathematics S2 2. Counting 2.3 Combinations Exercise9 : * 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? Casio enter n key : OPTN screenitem : PROB screenitem : ncr enter p key : EXE TI enter n key : MATH screenitem : PRB screenitem : ncr or Combinaisons enter p key : ENTER

I - Probability. What is Probability? the chance of an event occuring. 1classical probability. 2empirical probability. 3subjective probability

I - Probability. What is Probability? the chance of an event occuring. 1classical probability. 2empirical probability. 3subjective probability What is Probability? the chance of an event occuring eg 1classical probability 2empirical probability 3subjective probability Section 2 - Probability (1) Probability - Terminology random (probability)