05_Counting-1_packed

075 for n 94 fair bets

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Unformatted text preview: different! Ilya Pollak Birthday Problem: Solu8on •  Total number of outcomes: 365n. •  How many outcomes with all different birthdays? –  Birthday choices for 1st person: 365. –  Acceptable birthday choices for 2nd person: 364. –  Etc, total # of acceptable outcomes: 365 ·ȡ 364 ·ȡ … ·ȡ (366 – n) •  For n=2, this is 0.0027. •  For n=23, there is more than 50% chance to have two people with the same birthday. •  For n=71, the probability is 0.99932. Bet about 1.4¢ against $20. –  365!/(365- 71)!/365^71 on wolframalpha.com, then subtract result from 1. •  For n=94, the probability is 0.999998. Bet about 0.0039¢ against $20. Ilya Pollak P(at least two with the same birthday) as a func8on of n Full curve Zoomed in Ilya Pollak Example: Size of the power set of a finite set •  Power set of A is the set of all subsets of A. •  What is the number of subsets of {1,2,…,n}? •  Each element can either be included in a subset or not: 2 ·ȡ2 = 2n. 2 ·ȡ 2 ·ȡ … ·ȡ n times Ilya Pollak Birthday problem •  The earliest appearance seems to be in: •  W. Feller. An Introduc7on to Probability Theory and Its Applica7ons, 1950. •  [See Volume I, Chapter II, Sec8on 3 (page 33) of Third Edi8on, John Wiley & Sons, 1968.] Example: Playing dice •  What is the probability that six independent rolls of a fair six- sided die all give different numbers? •  Number of outcomes that make the event happen: 6 ·ȡ 5 ·ȡ 4 ·ȡ 3 ·ȡ 2 ·ȡ 1 = 6! = 720. •  Size of the sample space: 6 ·ȡ 6 ·ȡ 6 ·ȡ 6 ·ȡ 6 ·ȡ 6 = 66. •  Answer: 6!/66 = 5/324. Ilya Pollak Example: Combina8ons •  Combina8ons: k- element subsets of a given n- element set. •  Different from k- permuta8ons since there is no ordering. •  For one k- combina8on, k! different k- permuta8ons. This number is called “n choose k” An interes8ng consequence: Ilya Pollak Example: a set with 3 elements A Set S = B C Example: a set with 3 elements ⎛ 3⎞ number of zero-element subsets = ⎜ ⎟ ≡ 1---just the empty set ⎝0⎠ A B C ⎛ 3⎞ number of one-element subsets = ⎜ ⎟=3 ⎝1⎠ {A} , {B} , {C} Example: a set with 3 elements ⎛ 3⎞ number of zero-element subsets = ⎜ ⎟ ≡ 1---just the empty set ⎝0⎠ A B C ⎛ 3⎞ number of one-element subsets = ⎜ ⎟=3 ⎝1⎠ {A}...
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