Cut the Cards
08-21-2020, 11:00 PM
Post: #6
 Albert Chan Senior Member Posts: 1,890 Joined: Jul 2018
RE: Cut the Cards
(07-30-2020 08:58 PM)Albert Chan Wrote:  First card picked must never been seen, thus only 1 try needed to get 1st card.
Probability of getting the next "new" card = 51/52, or averaged 52/51 tries to get 2nd card
...

I might jumped too quickly to say averaged 1/p tries to get the next card.

Let probability of getting the card = p ; of not getting it = q = 1-p

Expected tries
= p + 2qp + 3q²p + 4q³p + ...
= (1-q) (1 + 2q + 3q² + 4q³ + ...)
= (1 + 2q + 3q² + 4q³ + ...) - (q + 2q² + 3q³ + ...)
= 1 + q + q² + q³ + ...
= 1/(1-q)
= 1/p

For n > 0, we have: ﻿ ﻿ ﻿ ﻿ $$\Large {1 \over (1-x)^n}\normalsize = \displaystyle \sum_{k=0}^∞ \binom{-n}{k}(-x)^k = \sum_{k=0}^∞ \binom{k+n-1}{n-1} x^k$$

Examples:

XCas> series(1/(1-x)^2, x, polynom) → 1+2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5
XCas> series(1/(1-x)^3, x, polynom) → 1+3*x + 6*x^2+10*x^3+15*x^4+21*x^5 ﻿ ﻿ // triangular numbers coefficients
XCas> series(1/(1-x)^4, x, polynom) → 1+4*x+10*x^2+20*x^3+35*x^4+56*x^5 ﻿ ﻿ // tetrahedral numbers coefficients
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 Messages In This Thread Cut the Cards - David Hayden - 07-30-2020, 08:00 PM RE: Cut the Cards - Albert Chan - 07-30-2020, 08:58 PM RE: Cut the Cards - Albert Chan - 08-21-2020 11:00 PM RE: Cut the Cards - Jim Horn - 07-30-2020, 09:49 PM RE: Cut the Cards - John Keith - 07-31-2020, 12:24 AM RE: Cut the Cards - Gerson W. Barbosa - 08-24-2020, 01:57 PM RE: Cut the Cards - Albert Chan - 08-25-2020, 06:14 PM RE: Cut the Cards - Albert Chan - 07-30-2020, 10:21 PM RE: Cut the Cards - pinkman - 08-24-2020, 09:49 PM RE: Cut the Cards - Gerson W. Barbosa - 08-25-2020, 11:41 PM RE: Cut the Cards - Albert Chan - 08-26-2020, 03:06 AM RE: Cut the Cards - Gerson W. Barbosa - 08-26-2020, 08:23 AM RE: Cut the Cards - Albert Chan - 08-26-2020, 02:13 PM RE: Cut the Cards - Gerson W. Barbosa - 08-26-2020, 06:13 PM RE: Cut the Cards - Gerson W. Barbosa - 08-27-2020, 10:07 PM RE: Cut the Cards - Albert Chan - 08-28-2020, 09:26 PM RE: Cut the Cards - Albert Chan - 08-29-2020, 04:02 PM RE: Cut the Cards - Gerson W. Barbosa - 08-28-2020, 11:39 PM RE: Cut the Cards - Albert Chan - 06-23-2021, 12:08 AM

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