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Combinatorics Jason Filippou CMSC250 @ UMCP 07-05-2016 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 1 / 42 Outline 1 The multiplication rule Permutations and combinations 2 The addition rule 3 Di ff erence rule 4 Inclusion /

  1. The multiplication rule Permutations and combinations Subsets of a set When talking about sets, order doesn’t matter! { 0 , 2 } = { 2 , 0 } Therefore, it doesn’t make sense to talk about permutations of a set ! Instead, an interesting question is how many subsets of a set are there. |P ( A ) | = 2 n , for | A | = n . We can prove this via induction, and by something known as the binomial theorem (which we might have time to talk about). Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 10 / 42

  2. The multiplication rule Permutations and combinations Subsets of a set When talking about sets, order doesn’t matter! { 0 , 2 } = { 2 , 0 } Therefore, it doesn’t make sense to talk about permutations of a set ! Instead, an interesting question is how many subsets of a set are there. |P ( A ) | = 2 n , for | A | = n . We can prove this via induction, and by something known as the binomial theorem (which we might have time to talk about). But how many subsets of size k ( k < n ) are there? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 10 / 42

  3. The multiplication rule Permutations and combinations Some examples We know that there are 6 substrings of size 2 for the string ”ACE” (discussed yesterday) How many subsets of size 2 of the set { A, C, E } are there? Something Else 3 6 8 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 11 / 42

  4. The multiplication rule Permutations and combinations Some examples We know that there are 6 substrings of size 2 for the string ”ACE” (discussed yesterday) How many subsets of size 2 of the set { A, C, E } are there? Something Else 3 6 8 If we have 5 students S 1 , S 2 , S 3 , S 4 , S 5 , how many pairs of students can we generate? P (5 , 2) d 5 / 2 e ! 5 2 5 ⇥ 2 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 11 / 42

  5. The multiplication rule Permutations and combinations Some examples We know that there are 6 substrings of size 2 for the string ”ACE” (discussed yesterday) How many subsets of size 2 of the set { A, C, E } are there? Something Else 3 6 8 If we have 5 students S 1 , S 2 , S 3 , S 4 , S 5 , how many pairs of students can we generate? P (5 , 2) d 5 / 2 e ! 5 2 5 ⇥ 2 If we have 5 students S 1 , S 2 , S 3 , S 4 , S 5 , how many ways can we pair them up in ? 10 ⇥ 2 5 P (10 , 2) 10 ⇥ 3 10 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 11 / 42

  6. The multiplication rule Permutations and combinations Combination formula Key question : Given a set of cardinality n , how many subsets of size r can I find? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 12 / 42

  7. The multiplication rule Permutations and combinations Combination formula Key question : Given a set of cardinality n , how many subsets of size r can I find? Call it X for now. Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 12 / 42

  8. The multiplication rule Permutations and combinations Combination formula Key question : Given a set of cardinality n , how many subsets of size r can I find? Call it X for now. Key observation: P ( n , r ) = X · r ! (Why?) Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 12 / 42

  9. The multiplication rule Permutations and combinations Combination formula Key question : Given a set of cardinality n , how many subsets of size r can I find? Call it X for now. Key observation: P ( n , r ) = X · r ! (Why?) Therefore, X = P ( n, r ) r ! We call X the number of r -combinations that we can choose from a set of n elements , and we symbolize it with C ( n, r ), or � n � ( “ n choose r ” ) . r Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 12 / 42

  10. The multiplication rule Permutations and combinations Notation! Definition (Number of r -combinations) Let n 2 N . Given a set with n elements, the number of � n � r -combinations that can be drawn from that set is symbolized as r (“ n choose r ”) and is equal to the formula: ✓ n ◆ = P ( n, r ) r r ! Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 13 / 42

  11. The multiplication rule Permutations and combinations Notation! Definition (Number of r -combinations) Let n 2 N . Given a set with n elements, the number of � n � r -combinations that can be drawn from that set is symbolized as r (“ n choose r ”) and is equal to the formula: ✓ n ◆ = P ( n, r ) r r ! Corollary (Factorial form of r -combination number) � n n ! � = r r !( n − r )! Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 13 / 42

  12. The multiplication rule Permutations and combinations Notation! Definition (Number of r -combinations) Let n 2 N . Given a set with n elements, the number of � n � r -combinations that can be drawn from that set is symbolized as r (“ n choose r ”) and is equal to the formula: ✓ n ◆ = P ( n, r ) r r ! Corollary (Factorial form of r -combination number) � n n ! � = r r !( n − r )! Corollary (Number of n -combinations) � n � = 1 n Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 13 / 42

  13. The multiplication rule Permutations and combinations Practice 100 people attend a cocktail party and everybody shakes hands with everybody else. How many handshakes occur? � 2 � 100 � 99 � � � P (99 , 2) 2 100 2 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 14 / 42

  14. The multiplication rule Permutations and combinations Practice 100 people attend a cocktail party and everybody shakes hands with everybody else. How many handshakes occur? � 2 � 100 � 99 � � � P (99 , 2) 2 100 2 Yesterday, I had you exchange your proofs by sitting 6 in each row, across 7 rows, and working in pairs. We fit the total number perfectly: 7 ⇥ 6 = 42 (43 students were attending, of which 1 helped me with a proof on the whiteboard). How many such exchanges were made? � 42 � 6 � 7 � � � 7 ⇥ 21 2 2 2 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 14 / 42

  15. The multiplication rule Permutations and combinations Practice 100 people attend a cocktail party and everybody shakes hands with everybody else. How many handshakes occur? � 2 � 100 � 99 � � � P (99 , 2) 2 100 2 Yesterday, I had you exchange your proofs by sitting 6 in each row, across 7 rows, and working in pairs. We fit the total number perfectly: 7 ⇥ 6 = 42 (43 students were attending, of which 1 helped me with a proof on the whiteboard). How many such exchanges were made? � 42 � 6 � 7 � � � 7 ⇥ 21 2 2 2 What if I had every possible pair of students in every row exchange proofs ? � 42 � 6 � 7 � � � 7 ⇥ 21 2 2 2 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 14 / 42

  16. The multiplication rule Permutations and combinations More practice I’m playing Texas Hold-Em poker and I’m sitting two positions on the left of the dealer, during the first hand of betting. Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 15 / 42

  17. The multiplication rule Permutations and combinations More practice I’m playing Texas Hold-Em poker and I’m sitting two positions on the left of the dealer, during the first hand of betting. How many di ff erent pairs of cards can the dealer deal to me, if he deals two cards at a time? � 52 � 50 � 48 � � � 2 5 2 2 2 2 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 15 / 42

  18. The addition rule The addition rule Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 16 / 42

  19. The addition rule Let’s solve a problem. Suppose we want to sign up for a website, and we are asked to create a password. Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 17 / 42

  20. The addition rule Let’s solve a problem. Suppose we want to sign up for a website, and we are asked to create a password. The website tells us: Your password, which should be at least 4 and at most 6 symbols long, must contain English lowercase or uppercase characters, digits, or one of the special characters #, ⇤ , , - , @ , & . Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 17 / 42

  21. The addition rule Let’s solve a problem. Suppose we want to sign up for a website, and we are asked to create a password. The website tells us: Your password, which should be at least 4 and at most 6 symbols long, must contain English lowercase or uppercase characters, digits, or one of the special characters #, ⇤ , , - , @ , & . How many passwords can I make? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 17 / 42

  22. The addition rule Let’s solve a problem. Suppose we want to sign up for a website, and we are asked to create a password. The website tells us: Your password, which should be at least 4 and at most 6 symbols long, must contain English lowercase or uppercase characters, digits, or one of the special characters #, ⇤ , , - , @ , & . How many passwords can I make? There are some passwords of length 4, N 4 = . . . Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 17 / 42

  23. The addition rule Let’s solve a problem. Suppose we want to sign up for a website, and we are asked to create a password. The website tells us: Your password, which should be at least 4 and at most 6 symbols long, must contain English lowercase or uppercase characters, digits, or one of the special characters #, ⇤ , , - , @ , & . How many passwords can I make? There are some passwords of length 4, N 4 = . . . There are some passwords of length 5, N 5 = . . . Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 17 / 42

  24. The addition rule Let’s solve a problem. Suppose we want to sign up for a website, and we are asked to create a password. The website tells us: Your password, which should be at least 4 and at most 6 symbols long, must contain English lowercase or uppercase characters, digits, or one of the special characters #, ⇤ , , - , @ , & . How many passwords can I make? There are some passwords of length 4, N 4 = . . . There are some passwords of length 5, N 5 = . . . And some of length 6, N 6 = . . . Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 17 / 42

  25. The addition rule Let’s solve a problem. Suppose we want to sign up for a website, and we are asked to create a password. The website tells us: Your password, which should be at least 4 and at most 6 symbols long, must contain English lowercase or uppercase characters, digits, or one of the special characters #, ⇤ , , - , @ , & . How many passwords can I make? There are some passwords of length 4, N 4 = . . . There are some passwords of length 5, N 5 = . . . And some of length 6, N 6 = . . . Final answer: N 4 + N 5 + N 6 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 17 / 42

  26. The addition rule The addition rule The addition rule actually has a set-theoretic base: Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 18 / 42

  27. The addition rule The addition rule The addition rule actually has a set-theoretic base: Theorem (Number of elements in disjoint sets) If A 1 , A 2 , . . . A n are finite, pairwise disjoint sets, then n n [ X | A i | = | A i | i =1 i =1 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 18 / 42

  28. The addition rule The addition rule The addition rule actually has a set-theoretic base: Theorem (Number of elements in disjoint sets) If A 1 , A 2 , . . . A n are finite, pairwise disjoint sets, then n n [ X | A i | = | A i | i =1 i =1 But in combinatorics, we only care to apply it when we have an experiment that can be split into discrete cases . Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 18 / 42

  29. The addition rule Practice! Number of substrings of length 4 and 5 built from English lowercase and uppercase characters, without repetitions. Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 19 / 42

  30. The addition rule Practice! Number of substrings of length 4 and 5 built from English lowercase and uppercase characters, without repetitions. I’m playing blackjack and I’m dealt 10  7 ♥ . How many di ff erent ways can I hit the blackjack (21)? ( Note: if you have more than one ace in your hand, one of them counts for 11) Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 19 / 42

  31. The addition rule Practice! Number of substrings of length 4 and 5 built from English lowercase and uppercase characters, without repetitions. I’m playing blackjack and I’m dealt 10  7 ♥ . How many di ff erent ways can I hit the blackjack (21)? ( Note: if you have more than one ace in your hand, one of them counts for 11) I’m playing blackjack and I’m dealt 10  7 ♥ . How many di ff erent ways can I hit the blackjack (21) in one hand ? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 19 / 42

  32. Di ff erence rule Di ff erence rule Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 20 / 42

  33. Di ff erence rule Subtraction / Di ff erence rule Definition (Di ff erence rule) If A , B are finite sets such that A ◆ B , | A � B | = | A | � | B | . Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 21 / 42

  34. Di ff erence rule Subtraction / Di ff erence rule Definition (Di ff erence rule) If A , B are finite sets such that A ◆ B , | A � B | = | A | � | B | . Let’s practice: 1 Of all 4-letter words in the English alphabet, how many do not begin with an ‘L’? 2 Of all straights in a deck, how many are not straight flushes? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 21 / 42

  35. Inclusion / Exclusion principle Inclusion / Exclusion principle Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 22 / 42

  36. Inclusion / Exclusion principle Definitions Again, has a set-theoretic base. Essentially is a corollary of the di ff erence rule. Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 23 / 42

  37. Inclusion / Exclusion principle Definitions Again, has a set-theoretic base. Essentially is a corollary of the di ff erence rule. Theorem (Inclusion-Exclusion principle for 2 sets) If A and B are finite sets, | A [ B | = | A | + | B | � | A \ B | Theorem (Inclusion-Exclusion principle for 3 sets) If A, B or C are finite sets, | A [ B [ C | = | A | + | B | + | C | � | A \ B | � | A \ C | � | B \ C | + | A \ B \ C | Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 23 / 42

  38. Inclusion / Exclusion principle Definitions Again, has a set-theoretic base. Essentially is a corollary of the di ff erence rule. Theorem (Inclusion-Exclusion principle for 2 sets) If A and B are finite sets, | A [ B | = | A | + | B | � | A \ B | Theorem (Inclusion-Exclusion principle for 3 sets) If A, B or C are finite sets, | A [ B [ C | = | A | + | B | + | C | � | A \ B | � | A \ C | � | B \ C | + | A \ B \ C | In our case: helps us calculate any one of the missing quantities given the other ones! Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 23 / 42

  39. Inclusion / Exclusion principle Practice! In a class of undergraduate Comp Sci students, 43 have taken 250, 52 have taken 216, while 20 have taken both. No student has taken any other courses. How many students are there? 63 95 75 72 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 24 / 42

  40. Inclusion / Exclusion principle Practice! In a class of undergraduate Comp Sci students, 43 have taken 250, 52 have taken 216, while 20 have taken both. No student has taken any other courses. How many students are there? 63 95 75 72 I have 25 players in a school football team and I want to find versatile players, that can play all positions. 3 can play as halfbacks, 13 can play as fullbacks, while 6 can play as tight ends. 8 can play as both halfbacks and fullbacks. 4 can play as both tight ends and fullbacks, while 2 can play as tight ends and halfbacks. How many players can play all 3 positions? 1 14 24 16 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 24 / 42

  41. Inclusion / Exclusion principle Practice! In a class of undergraduate Comp Sci students, 43 have taken 250, 52 have taken 216, while 20 have taken both. No student has taken any other courses. How many students are there? 63 95 75 72 I have 25 players in a school football team and I want to find versatile players, that can play all positions. 3 can play as halfbacks, 13 can play as fullbacks, while 6 can play as tight ends. 8 can play as both halfbacks and fullbacks. 4 can play as both tight ends and fullbacks, while 2 can play as tight ends and halfbacks. How many players can play all 3 positions? 1 14 24 16 I’m playing poker, and I have been dealt 2 ♦ 5 ♦ preflop (at the first round of betting, before any community cards are dealt). In how many ways can I flop a flush, but not a straight flush ? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 24 / 42

  42. Inclusion / Exclusion principle Visualizing the multiplication and addition rules A very intuitive way to visualize the multiplication and addition rules is the possibility tree . Levels of the tree represent steps of our experiment. We branch o ff to all di ff erent ways to complete the next step. Example (clothes choices): White or blue fedora White, red, or black shirt Brown or black slacks Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 25 / 42

  43. Inclusion / Exclusion principle Visualizing the multiplication and addition rules Total of 12 outcomes (# leaves) But suppose that we do not allow for some combinations, e.g a red shirt will never work with brown slacks, and a blue fedora will never work with a red shirt. How do we work then? Constrain the tree! (Erase subtrees) Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 26 / 42

  44. Probabilities Probabilities Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 27 / 42

  45. Probabilities Definitions Definition (Sample Space) A sample space Ω is the set of all possible outcomes of an experiment. Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 28 / 42

  46. Probabilities Definitions Definition (Sample Space) A sample space Ω is the set of all possible outcomes of an experiment. Examples: 1 Tossing a coin: Ω = { H, T } Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 28 / 42

  47. Probabilities Definitions Definition (Sample Space) A sample space Ω is the set of all possible outcomes of an experiment. Examples: 1 Tossing a coin: Ω = { H, T } 2 Tossing two coins one after the other: Ω = { HH, HT, TH, TT } Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 28 / 42

  48. Probabilities Definitions Definition (Sample Space) A sample space Ω is the set of all possible outcomes of an experiment. Examples: 1 Tossing a coin: Ω = { H, T } 2 Tossing two coins one after the other: Ω = { HH, HT, TH, TT } 3 Grade in 250: { A, B, C, D, W, F, XF } Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 28 / 42

  49. Probabilities Definitions Definition (Sample Space) A sample space Ω is the set of all possible outcomes of an experiment. Examples: 1 Tossing a coin: Ω = { H, T } 2 Tossing two coins one after the other: Ω = { HH, HT, TH, TT } 3 Grade in 250: { A, B, C, D, W, F, XF } Definition (Event) Let Ω be a sample space. Then, any subset of Ω is called an event . Examples (corresponding to the above sample spaces) Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 28 / 42

  50. Probabilities Definitions Definition (Sample Space) A sample space Ω is the set of all possible outcomes of an experiment. Examples: 1 Tossing a coin: Ω = { H, T } 2 Tossing two coins one after the other: Ω = { HH, HT, TH, TT } 3 Grade in 250: { A, B, C, D, W, F, XF } Definition (Event) Let Ω be a sample space. Then, any subset of Ω is called an event . Examples (corresponding to the above sample spaces) { T } (Tails) Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 28 / 42

  51. Probabilities Definitions Definition (Sample Space) A sample space Ω is the set of all possible outcomes of an experiment. Examples: 1 Tossing a coin: Ω = { H, T } 2 Tossing two coins one after the other: Ω = { HH, HT, TH, TT } 3 Grade in 250: { A, B, C, D, W, F, XF } Definition (Event) Let Ω be a sample space. Then, any subset of Ω is called an event . Examples (corresponding to the above sample spaces) { T } (Tails) { HH, HT, TH } (How would you name this event?) Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 28 / 42

  52. Probabilities Definitions Definition (Sample Space) A sample space Ω is the set of all possible outcomes of an experiment. Examples: 1 Tossing a coin: Ω = { H, T } 2 Tossing two coins one after the other: Ω = { HH, HT, TH, TT } 3 Grade in 250: { A, B, C, D, W, F, XF } Definition (Event) Let Ω be a sample space. Then, any subset of Ω is called an event . Examples (corresponding to the above sample spaces) { T } (Tails) { HH, HT, TH } (How would you name this event?) { A, B, C } (A student passes 250) Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 28 / 42

  53. Probabilities Probability Definition (Probability of equally likely outcomes) Let Ω be a finite sample space where all outcomes are equally likely to occur and E be an event. Then, P ( E ) = | E | | Ω | . Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 29 / 42

  54. Probabilities Probability Definition (Probability of equally likely outcomes) Let Ω be a finite sample space where all outcomes are equally likely to occur and E be an event. Then, P ( E ) = | E | | Ω | . How plausible is it that all outcomes in a sample space are equally likely to occur in practice? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 29 / 42

  55. Probabilities Practice with sample spaces and probability Examples of certain experiments: 1 I toss the same coin 3 times. Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 30 / 42

  56. Probabilities Practice with sample spaces and probability Examples of certain experiments: 1 I toss the same coin 3 times. What’s my sample space Ω ? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 30 / 42

  57. Probabilities Practice with sample spaces and probability Examples of certain experiments: 1 I toss the same coin 3 times. What’s my sample space Ω ? What’s the size of my sample space? ( | Ω | ). Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 30 / 42

  58. Probabilities Practice with sample spaces and probability Examples of certain experiments: 1 I toss the same coin 3 times. What’s my sample space Ω ? What’s the size of my sample space? ( | Ω | ). What’s the probability that I don’t get any heads? 1 1 1 Something else 3 8 9 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 30 / 42

  59. Probabilities Practice with sample spaces and probability Examples of certain experiments: 1 I toss the same coin 3 times. What’s my sample space Ω ? What’s the size of my sample space? ( | Ω | ). What’s the probability that I don’t get any heads? 1 1 1 Something else 3 8 9 2 I roll two dice. Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 30 / 42

  60. Probabilities Practice with sample spaces and probability Examples of certain experiments: 1 I toss the same coin 3 times. What’s my sample space Ω ? What’s the size of my sample space? ( | Ω | ). What’s the probability that I don’t get any heads? 1 1 1 Something else 3 8 9 2 I roll two dice. What’s the sample space? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 30 / 42

  61. Probabilities Practice with sample spaces and probability Examples of certain experiments: 1 I toss the same coin 3 times. What’s my sample space Ω ? What’s the size of my sample space? ( | Ω | ). What’s the probability that I don’t get any heads? 1 1 1 Something else 3 8 9 2 I roll two dice. What’s the sample space? What’s the size of the sample space? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 30 / 42

  62. Probabilities Practice with sample spaces and probability Examples of certain experiments: 1 I toss the same coin 3 times. What’s my sample space Ω ? What’s the size of my sample space? ( | Ω | ). What’s the probability that I don’t get any heads? 1 1 1 Something else 3 8 9 2 I roll two dice. What’s the sample space? What’s the size of the sample space? What’s the probability that I hit 7? 1 1 7 Something else 12 6 12 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 30 / 42

  63. Probabilities Practice with sample spaces and probability Examples of certain experiments: 1 I toss the same coin 3 times. What’s my sample space Ω ? What’s the size of my sample space? ( | Ω | ). What’s the probability that I don’t get any heads? 1 1 1 Something else 3 8 9 2 I roll two dice. What’s the sample space? What’s the size of the sample space? What’s the probability that I hit 7? 1 1 7 Something else 12 6 12 3 I uniformly select a real number r between 0 and 10 inclusive. Sample space? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 30 / 42

  64. Probabilities Practice with sample spaces and probability Examples of certain experiments: 1 I toss the same coin 3 times. What’s my sample space Ω ? What’s the size of my sample space? ( | Ω | ). What’s the probability that I don’t get any heads? 1 1 1 Something else 3 8 9 2 I roll two dice. What’s the sample space? What’s the size of the sample space? What’s the probability that I hit 7? 1 1 7 Something else 12 6 12 3 I uniformly select a real number r between 0 and 10 inclusive. Sample space? Size of the sample space? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 30 / 42

  65. Probabilities Practice with sample spaces and probability Examples of certain experiments: 1 I toss the same coin 3 times. What’s my sample space Ω ? What’s the size of my sample space? ( | Ω | ). What’s the probability that I don’t get any heads? 1 1 1 Something else 3 8 9 2 I roll two dice. What’s the sample space? What’s the size of the sample space? What’s the probability that I hit 7? 1 1 7 Something else 12 6 12 3 I uniformly select a real number r between 0 and 10 inclusive. Sample space? Size of the sample space? Probability that r ∈ [4 , 5]? 1 1 Something else 0 10 9 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 30 / 42

  66. Probabilities NBA Playo ff s and shifting probabilities Figure 1: NBA playo ff s, 2012 Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 31 / 42

  67. Probabilities Some poker examples We’ve been dealt 2  4  . What is the probability that we are dealt a flush? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 32 / 42

  68. Probabilities Some poker examples We’ve been dealt 2  4  . What is the probability that we are dealt a flush? We’ve been dealt 2  4  . What is the probability that we are dealt a straight? Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 32 / 42

  69. Probabilities Some poker examples We’ve been dealt 2  4  . What is the probability that we are dealt a flush? We’ve been dealt 2  4  . What is the probability that we are dealt a straight? Your homework examples! (Let’s look them up again) Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 32 / 42

  70. Probabilities Joint, disjoint, dependent, independent events Joint, disjoint, dependent, independent events Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 33 / 42

  71. Probabilities Joint, disjoint, dependent, independent events The general case We know, via inclusion - exclusion principle, that the following holds: A [ B = A + B � A \ B (1) Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 34 / 42

  72. Probabilities Joint, disjoint, dependent, independent events The general case We know, via inclusion - exclusion principle, that the following holds: A [ B = A + B � A \ B (1) By the definition of probability, P ( E ) = | E | | Ω | (2) Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 34 / 42

  73. Probabilities Joint, disjoint, dependent, independent events The general case We know, via inclusion - exclusion principle, that the following holds: A [ B = A + B � A \ B (1) By the definition of probability, P ( E ) = | E | | Ω | (2) By (1) and (2) we have: P ( A [ B ) = P ( A ) + P ( B ) � P ( A \ B ) All discussion of disjoint and independent events begins from here. Jason Filippou (CMSC250 @ UMCP) Combinatorics 07-05-2016 34 / 42

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