Re: A card game probability

In article
<1169756728.912704.141420@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
,
"jseppa17@xxxxxxxxx" <jseppa17@xxxxxxxxx> wrote:

Take a card deck with 52 cards. Pick cards one by one and compute the
cards by ace, two, three,..., jack, queen, king, ace,..., king, ace,
..., king, ace, ..., king. What is the probability that at least once
you turn a card of the same value as you say aloud? Here ace=1,
jack=11, queen=12, king=13.

This distribution can be calculated using rook polynomials,
and a bit of computer algebra. The theory is in
Riordan, An_Introduction_to_Combinatorial_Analysis, Chapter 7.

p(X) = 1 + 16*X + 72*X^2 + 96*X^3 + 24*X^4
enumerates the number of ways of putting 0, 1, 2, 3, 4
non-taking rooks on a 4x4 board. We want to enumerate
the number of was of placing 52 non-taking rooks on a
52x52 board with restricted position. The restricted
positions are 13 4x4 squares on the main diagonal. The
rook polynomial for this board is
R(X) = (p(X))^13 = sum_k r_k * X^k.
Using the inclusion-exclusion principal the hit polynomial is
N(t) = sum_k r_k * (n - k)! * (t - 1)^k.
The probability generating function is N(t) / 52!.

The probability that there are no hits is the
coefficient of t^0. The probability that there is one
or more hits is
0.983767272532805363251870443486721
The exact value is
1 - 4610507544750288132457667562311567997623087869
/ 284025438982318025793544200005777916187500000000

N(t) / 52! =
0.01623272746719463674812955651*t^0
+ 0.06889908141478148238480010616*t^1
+ 0.1441638302112860873268198470*t^2
+ 0.1981941697388588924094429962*t^3
+ 0.2013248802631002121712218224*t^4
+ 0.1611121580673676374964015045*t^5
+ 0.1057624399417951088893824209*t^6
+ 0.05855394190643602745517394869*t^7
+ 0.02789763723028095706332388840*t^8
+ 0.01161476075337287702962956282*t^9
+ 0.004276419853299269025834876972*t^10
+ 0.001405846625821968214438444328*t^11
+ 0.0004158889042213801269282777817*t^12
+ 0.0001114318645215406764758314345*t^13
+ 0.00002718879450159080381337660546*t^14
+ 0.000006068934168765392501699277176*t^15
+ 0.000001244160563424403106461712668*t^16
+ 0.0000002350387861121421393840030417*t^17
+ 0.00000004103524651132255590270235256*t^18
+ 0.000000006637629910356887260777381124*t^19
+ 0.0000000009968844920903189737108741970*t^20
+ 1.392716324517475873638747498 E-10*t^21
+ 1.812864649192270581797705397 E-11*t^22
+ 2.201675891283604268999809850 E-12*t^23
+ 2.497693788550700554188668036 E-13*t^24
+ 2.649452660966172670988703214 E-14*t^25
+ 2.630061096624977268699657496 E-15*t^26
+ 2.444917118994548458919937732 E-16*t^27
+ 2.129544788140638742433241381 E-17*t^28
+ 1.738657537272995577404480295 E-18*t^29
+ 1.330991786333218416915551989 E-19*t^30
+ 9.555509373432967847634876975 E-21*t^31
+ 6.434147741117614051093241934 E-22*t^32
+ 4.063427645342916623890867709 E-23*t^33
+ 2.406769790338065887617888187 E-24*t^34
+ 1.336816981101869502840349571 E-25*t^35
+ 6.962226359032889587085380790 E-27*t^36
+ 3.399436902620704900228185016 E-28*t^37
+ 1.555990030849186628556731006 E-29*t^38
+ 6.676301916788635363743876401 E-31*t^39
+ 2.685524513147643440007637576 E-32*t^40
+ 1.012937605371499172949089347 E-33*t^41
+ 3.584000683866892746428521734 E-35*t^42
+ 1.190249033309115622119087605 E-36*t^43
+ 3.713185759212891448612512053 E-38*t^44
+ 1.088988664761155514091063686 E-39*t^45
+ 3.010131622783199448791785964 E-41*t^46
+ 7.787095906824826920876645351 E-43*t^47
+ 1.972202058951585606005723006 E-44*t^48
+ 3.978082200165939678608758800 E-46*t^49
+ 1.356164386420206708616622318 E-47*t^50
+ 0.0 *t^51
+ 1.086670181426447683186396088 E-50*t^52

--
Michael Press
.