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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 zeroelement subsets = ⎜
⎟ ≡ 1just the empty set
⎝0⎠ A
B C ⎛ 3⎞
number of oneelement subsets = ⎜
⎟=3
⎝1⎠ {A} , {B} , {C} Example: a set with 3 elements ⎛ 3⎞
number of zeroelement subsets = ⎜
⎟ ≡ 1just the empty set
⎝0⎠ A
B C ⎛ 3⎞
number of oneelement subsets = ⎜
⎟=3
⎝1⎠ {A}...
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 Fall '08
 GELFAND
 Electrical Engineering

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