⇐ Goals: Probability CS 70 Tips The probability section in CS 70 usually means: I Lets you quantify uncertainty I One big topic , rather than many small topics I Concretely: has applications everywhere! I Try your best to stay up to date ; use OH! Counting, Part I I Important to be comfortable with the basics I Hopefully: learn techniques for reasoning about randomness and making better decisions logically I Fewer “proofs,” more computations CS 70, Summer 2019 I Emphasis on applying tools and problem solving I Hopefully: provides a new perspective on the world I Lectures will be example-driven Lecture 13, 7/16/19 I Practice, practice, practice! 1 / 29 2 / 29 3 / 29 A Familiar Question Choices, Choices, Choices... The First Rule of Counting: Products How many bit (0 or 1) strings are there of length 3? A lunch special lets you choose one appetizer, one entre´ e, and If the object you are counting: one drink. There are 6 appetizers, 3 entre´ es, and 5 drinks. 3 choices I Comes from making k choices How many di ff erent meals could you possibly get? I Has n 1 options for the first choice -¥ ¥ tf Drink App Ent I Has n 2 options for second, regardless of the first bit I Has n 3 options for the third, regardless of the first two I ...and so on, until the k -th choice ¥÷ = ⇒ Count the object using the product n 1 × n 2 × n 3 × . . . × n k EE . ⑧ 90 4 / 29 5 / 29 6 / 29
⇒ Anagramming I Counting Functions Counting Polynomials ' in : " Exactly How many strings can we make by rearranging “CS70”? How many functions are there from { 1 , . . . , n } to { 1 , . . . , m } ? How many degree d polynomials are there modulo p ? ÷ 4×3×2×1=24 ' - Dpd Pia '¥ =p ⇐ ix. character T : - - - . . f ( N ) # 4 . . . # 2 # 3 # ÷ ④ = MN - 17cm -27 2-1 non = - - - O How many strings can we make by rearranging “ILOVECS70” if Same setup, but m ≥ n . How many injective functions are there? If d ≤ p , how many have no repeating coe ffi cients? . . . the numbers “70” must appear together in that order? pretend 't 's Exercise . " ¥ ' m 1 letter . -8×-71 I -8 ! characters : - - - - - - ND ! ( M 7 / 29 8 / 29 9 / 29 When Order Doesn’t Matter: Space Team I When Order Doesn’t Matter: Poker I The Second Rule of Counting: Repetitions Among its 10 trainees, NASA wants to choose 3 to go to the In poker, each player is dealt 5 cards. A standard deck (no jokers) Say we use the First Rule–we make k choices. lot O moon. How many ways can they do this? has 52 cards. How many di ff erent hands could you get? I Let A be the set of ordered objects. = 7 ! 8=720 I Let B be the set of unordered objects. × 9 10 x 48=51 ¥3 people ¥ 52 51 It SO 49 If there is an “ m -to- 1 ” function from A to B : cards 47 ! - - - - - BAC } = ⇒ Count A and divide by m to get | B | . ABC ¥93 7261=120 ACB ! se !?¥}s - repetitions 6 retorting offs ,z , 47 ! 5 ! Yt EEE =3 ! - - - CBA 10 / 29 11 / 29 12 / 29
⇒ ⇒ Anagramming II Binomial Coe ffi cients Binomial Coe ffi cients How many strings can we make by rearranging “APPLE”? How many ways can we... We often use 12 � n � n ! LE } AP = = { anagrams of , Pa 5 ! I pick a set of 2 items out of n total? A . k k !( n − k )! - D # I APPLE ftp.kp.EE > M - -2 J to represent the number of ways to choose k out of n items when M : reps a m : . order doesn’t matter. cnn.IM?N-InDxIn-2 - -521=60 Hmt (B) . I pick a set of 3 items out of n total? How many strings can we make by rearranging “BANANA”? I 2 23 ) 1 ni 6 ! y IAI We call this quantity “ n choose k ” . - - 3 ! reps - 3) ! 3 ! th We also sometimes refer to these as “binomial coe ffi cients.” 2 ! :3 ! A 's N 's : M : 243 ! I pick a set of k items out of n total? Total : Mt ⇒ tAmt=2÷ Q: Using this definition, what does 0 ! equal? IBI - K ) ! K ! ( n . 13 / 29 14 / 29 15 / 29 Binomial Coe ffi cients Anagramming III Coincidence? - - K O 'S Using this definition, what does 0 ! equal? How many bit strings can we make by k 1’s and ( n − k ) 0’s? Is there a relationship between: an ( not 't 1. Length n bit strings with k 1’s, and n? oh ' . IKO . On 1,12 , Oz 2. Ways of choosing k items from n when order doesn’t matter? = K . . - . . - - . N NO ! ! aO - - K K n Yes! n ! � n with → ordering : - � n - - � � - - Should we be surprised that = ? k ! . - K X 's k n − k I 's > k ! ( n - K ) ! X Repetitions : X : X O O - K n ! O 's n ! n - k ) ! O 's ( n - : K ) ! K ! - item - - xlo In strings t k b/w k ! ( n - K ) ! - bijection ⇒ Yin sets - Ik ) in in T - k members 2/0 - of choosing ways NOT ON members team ! K my team For 16 / 29 17 / 29 18 / 29
⇒ ⇒ ⇒ Putting It All Together: Space Team II Putting It All Together: Poker II ④ Exercise Sampling Without Replacement Among its 10 trainees, NASA wants to choose 3 to go to the How many 5-card poker hands form a full house (triple + pair)? How many ways can we sample k items out of n items, without moon, and 2 to go to Mars. They also don’t want anyone to do replacement , if: both missions. How many ways can they choose teams? I Order matters? Is Ii NIN - ktl ) - 1) Hn - 2) r In -= - K ) ! - - - - - How many 5-card poker hands form a straight (consecutive - in - items - cards), including straight flushes (same suit)? K If one member of the moon mission is designated as a captain, I Order does not matter? how many ways can they choose teams? ④ counting ) " ( second of er rule E) 6 i¥ How many 5-card poker hands form two pairs? ( YL ) k ! orderings per set Eoin . Tars We were able to use the First and Second rules of counting! 19 / 29 20 / 29 21 / 29 Sampling With Replacement When Repetitions Aren’t Uniform: Splitting Money When Repetitions Aren’t Uniform: Splitting Money How many ways can we sample k items out of n total items, with Alice, Bob, and Charlie want to split $6 amongst themselves. Second attempt: the “divider” point of view replacement , if: O First (naive and di ffi cult) attempt: the “dollar’s point of view” ¥ffBob#azµ I Order matters? -3×3-+3×-3=36 , ¥3 -_ Mk o , 00→⑧A08⑧#z • I X -N×N_ pg items Dollar orderndoesnt#ter . . ④ . o 's 6 - are dollars because indistinguishable K oos.o.EE#80zNsswpfiiYhng$6 . items . ,N Ai ice b Label I I Order does not matter? , rearranging . . . ABAAAAA# Iufagyet . , I } { I I I 111 , , . . . . . AAAAA qq.to#ARSANBARSf(8T)=&2. . ,24 { 1,1 9%5 I ? AAA } , . . repetitions b BA Ek Yoaysget 7. 2nd ! rule ④ CANNOT Use What can we do when order does not matter? 22 / 29 23 / 29 24 / 29
↳ “Stars and Bars” Application: Sums to k Summary Pick Your Strategy I How many ways can we choose n (not necessarily distinct) You have 12 distinct cards and 3 people. How many ways to: I k choices, always the same number of options at choice i " with non-negative numbers that sum to k ? " replace I Deal to the 3 people in sequence (4 cards each), and the regardless of previous outcome = ⇒ First Rule set negative xi order they received the cards matters? . 's TINI I Order doesn’t matter; same number of repetitions for each - incest . . desired outcome = ⇒ Second Rule f :p - ites I Indistinguishable items split among a fixed number of Yn - ! @ di ff erent buckets = ⇒ Stars and Bars k! I Deal to the 3 people in sequence (4 cards each), but order integers doesn’t matter? Food for thought: What if the numbers have to be positive ? to Xz Xs Xn • of .co/.of.oofFYnBo' § Xi . . . o , ¥¥÷ : " " Isao . 25 / 29 26 / 29 27 / 29 Pick Your Strategy II Pick Your Strategy III You have 12 distinct cards and 3 people. How many ways to: There are n citizens on 5 different committees. Say n > 15, and that each citizen is on at most 1 committee. I Deal 3 piles in sequence (4 cards each), and don’t distinguish How many ways to: the piles? I Assign a leader to each committee, then distribute all n − 5 remaining citizens in any way? I The cards are now indistinguishable. How many ways to deal so that each person receives at least 2 cards? I Assign a captain and two members to each committee? 28 / 29 29 / 29
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