(11C) Probability of No Repetitions

01102020, 12:42 PM
(This post was last modified: 01112020 01:02 AM by Gamo.)
Post: #1




(11C) Probability of No Repetitions
This program was adapted from HP55 Statistic Book (Page 12)
Reference: E. Parzen, Modern Probability Theory and its Applications, John Wiley and Sons, 1960 (CH. 2 Page 46) As stated in the book: Probability of No Repetitions in a Sample Suppose a sample of size n is drawn with replacement from population containing m different objects. Let P be the probability that there are no repetitions in the sample, then P = [1 (1/m)][1 (2/m)]....[1 (n1/m)] Given integer m, n such that m ≥ n ≥ 1 this program finds the probability P. Remark: The execution time of the program depends on n; the larger n is, the longer it takes.  Example: HP55 Statistic Book page 13 In a room containing n persons, what is the probability that no two or more persons have the same birthday for n = 4, 23, 48? Note: m = 365 // 365 is the days amount of birthday [USER] [FIX] 2 365 [A] display 365.00 // Enter m 4 [B] display brieftly 0.98 then 2.00 // Enter n 23 [B] display briefly 0.49 the 51.00 48 [B] display briefly 0.04 then 96.00 Answer: In a room for the probability that at lease two of them will have the same birthday 4 people in a room P = 0.98 or only 2% 23 people in a room P = 0.49 or 51% 48 people in a room P = 0.04 or 96%  Program: Code:
Gamo 1/2020 

01112020, 01:46 AM
(This post was last modified: 01112020 09:02 PM by Albert Chan.)
Post: #2




RE: (11C) Probability of No Repetitions
(01102020 12:42 PM)Gamo Wrote: Probability of No Repetitions in a Sample This is similar to thread (42S) Probability of Same Birthday Day We can do approximation the same way: products of terms = exp of sum of ln(term) Sum is approximated by integral with correction (bolded expression below) If we let u = 1x/m, we have x = mm*u XCas> f := ln(1x/m) XCas> expand(subst(int(f)  f/2 + diff(f)/12, x = mm*u)) → m*u  1/(12*m*u)  ln(u)/2  m*u*ln(u) XCas> g(m,u) := 1/(12*m*u)  ln(u)*(0.5 + m*u) + m*u // ≈ sum formula XCas> p(m,n) := exp(g(m, 1.n/m)  g(m, 1.1/m)) // ≈ product(1x/m, x=1 .. n1) XCas> p(365, 4) → 0.9836440875 XCas> p(365,23) → 0.4927027657 XCas> p(365,48) → 0.03940202712 

01112020, 02:15 PM
(This post was last modified: 01132020 01:05 PM by Albert Chan.)
Post: #3




RE: (11C) Probability of No Repetitions
(01102020 12:42 PM)Gamo Wrote: P = [1 (1/m)][1 (2/m)]....[1 (n1)/m] Perhaps it is more useful by adding a (10/m) term, we have P = perm(m,n) / m^{n} Using the same idea to approximate ln(perm(m,n)): XCas> f := ln(mx) XCas> expand(subst(int(f)  f/2 + diff(f)/12, x = mu)) → 1/(12*u)  ln(u)/2  u*ln(u) + u XCaS> g(u) := 1/(12*u)  ln(u)/2  u*ln(u) + u XCas> ln_perm(m,n) := g(mn)  g(m) // integral limit x = 0 to n, u=mx ln_perm() can be simplified and more accurate by putting tiny terms up front, and use logp1() \(\large \ln(perm(m,n)) ≈ {n\over 12m(mn)}  [n + (mn+½)\ln(1  n/m)]+ n \ln(m) \) For ln(P), above last term cancelled out. XCas> ln_P(m,n) := n/(12*m*(mn))  (n + (mn+0.5)*ln(1n/m)) XCas> ln_perm(m,n) := ln_P(m,n) + n*ln(m) XCas> e^ln_P(365, 4) → 0.9836440875 XCas> e^ln_P(365,23) → 0.4927027657 XCas> e^ln_P(365,48) → 0.03940202712 Trivia: ln_perm(m,n) = ln_perm(m,a) + ln_perm(ma, na), same as real permutation function. if m ≈ n, ln_perm(m,n) is not very accurate. With above trivia, splitting ln_perm(m,n) does not help. However, we can replace the 2nd term with real ln(perm(ma,na)) Example: ln(perm(69,68)) = ln(69!) ≈ 226.1905483 XCas> ln_perm(69,68) → 226.1882765 // slightly worse than log of Stirling's formula. XCas> ln_perm(68,67) + ln(2) → 226.1902224 // perm(6967, 6867) = perm(2,1) = 2 XCas> ln_perm(68,66) + ln(6) → 226.1904485 // perm(6966, 6866) = perm(3,2) = 6 

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