TC Mathematics S2 Coins Die dice Tale Head Set/deck of playing cards Spades Hearts Diamonds Clubs
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 12096/201376 = 54/899 0.06
TC Mathematics S2 Sets Count PROBABILITIES
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
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)
TC Mathematics S2 1. Counting 1.1 Reminders on sets A 2 4 6 8 10 12 1 5 3 7 11 9 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 = 1 3 5 6 7 9 11 12
TC Mathematics S2 1. Counting 1.1 Reminders on sets Properties of operations
TC Mathematics S2 1. Counting 1.1 Reminders on sets Properties of operations A 2 4 6 8 10 12 1 5 3 7 11 9 E B
TC Mathematics S2 1. Counting 1.1 Reminders on sets Cardinal number of a set A 2 4 6 8 10 12 1 5 3 7 11 9 E B Contingency table
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)
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
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
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)
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)
TC Mathematics S2 0 2. 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 8 5 2 0 p i p(a) p(b) p(c) p(d)
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, }:# =?
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
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)
TC Mathematics S2 2.1 p-lists 1st figure 2. Counting 1 2 3 4 5 Let sdrawa CHOICE TREE : 2nd figure 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 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
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}?
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»
TC Mathematics S2 2. Counting 2.2 Permutations Let sdrawa CHOICE TREE : 1st figure 1 2 3 4 5 2nd figure 2 3 4 5 1 3 4 5 1 2 4 5 1 2 3 5 1 2 3 4 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
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
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
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»
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
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? * 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