perplexus.info :: Probability : 52 Pickup

52 Pickup (Posted on 2008-01-15)

52-pickup is the practical joke "game" where the mark finds out that the game consists of the proposer spraying all 52 cards of a deck into the air, letting them fall onto the floor, and then the mark has to pick them up.

Suppose all 52 cards are now on the floor, each card randomly and independently face up or down with probability 1/2.

What is the probability that the sum of the face-up cards (counting A, J, Q, K as 1, 11, 12, 13 respectively) is a multiple of 13?

Submitted by Charlie

Rating: 5.0000 (1 votes)

Solution: (Hide)

The states of the Kings don't count. The following program counts the probability of each of the 12 other possible values, adding mod 13, for each stage of counting Aces, deuces, etc. with probability 1/16, 1/4, 3/8, 1/4 and 1/6 of getting 0 through 4 face-up of that denomination, and considers adding, mod 13, that many times the probability of each, to contribute to the probability of that augmented value, before considering the next denomination:

10 dim Prob(12),TransP(4),Chg(4),NewProb(12) 20 Prob(0)=1 30 TransP(0)=1//16:TransP(1)=1//4:TransP(2)=6//16:TransP(3)=1//4:TransP(4)=1//16 40 for Denom=1 to 12 50 for I=0 to 4 60 Chg(I)=(Denom*I)@13 70 next 80 erase NewProb() 90 dim NewProb(12) 100 for I=0 to 12 110 for J=0 to 4 120 V=(I+Chg(J))@13 130 NewProb(V)=NewProb(V)+Prob(I)*TransP(J) 140 next J 150 next I 155 for I=0 to 12:Prob(I)=NewProb(I):next 160 next Denom 170 print Prob(0),1/Prob(0)

The result is 5412980321359/70368744177664 ~= 1/12.9999999999994457766
The table below shows the probabilities of the values of the total mod 13, first after all the showing aces are counted, then after the aces and deuces, etc.

After 1's 0 1/16 = 0.0625 1 1/4 = 0.25 2 3/8 = 0.375 3 1/4 = 0.25 4 1/16 = 0.0625 5 0 = 0 6 0 = 0 7 0 = 0 8 0 = 0 9 0 = 0 10 0 = 0 11 0 = 0 12 0 = 0 After 2's 0 1/256 = 0.00390625 1 1/64 = 0.015625 2 5/128 = 0.0390625 3 5/64 = 0.078125 4 31/256 = 0.12109375 5 5/32 = 0.15625 6 11/64 = 0.171875 7 5/32 = 0.15625 8 31/256 = 0.12109375 9 5/64 = 0.078125 10 5/128 = 0.0390625 11 1/64 = 0.015625 12 1/256 = 0.00390625 After 3's 0 409/4096 = 0.099853515625 1 47/512 = 0.091796875 2 165/2048 = 0.08056640625 3 275/4096 = 0.067138671875 4 235/4096 = 0.057373046875 5 105/2048 = 0.05126953125 6 105/2048 = 0.05126953125 7 235/4096 = 0.057373046875 8 275/4096 = 0.067138671875 9 165/2048 = 0.08056640625 10 47/512 = 0.091796875 11 409/4096 = 0.099853515625 12 213/2048 = 0.10400390625 After 4's 0 4869/65536 = 0.0742950439453125 1 4869/65536 = 0.0742950439453125 2 2451/32768 = 0.074798583984375 3 2489/32768 = 0.075958251953125 4 5067/65536 = 0.0773162841796875 5 1285/16384 = 0.07843017578125 6 5199/65536 = 0.0793304443359375 7 2613/32768 = 0.079742431640625 8 5199/65536 = 0.0793304443359375 9 1285/16384 = 0.07843017578125 10 5067/65536 = 0.0773162841796875 11 2489/32768 = 0.075958251953125 12 2451/32768 = 0.074798583984375 After 5's 0 20161/262144 = 0.076908111572265625 1 80665/1048576 = 0.0769281387329101562 2 80685/1048576 = 0.0769472122192382812 3 20175/262144 = 0.076961517333984375 4 40353/524288 = 0.0769672393798828125 5 20175/262144 = 0.076961517333984375 6 80685/1048576 = 0.0769472122192382812 7 80665/1048576 = 0.0769281387329101562 8 20161/262144 = 0.076908111572265625 9 40313/524288 = 0.0768909454345703125 10 80615/1048576 = 0.0768804550170898437 11 80615/1048576 = 0.0768804550170898437 12 40313/524288 = 0.0768909454345703125 After 6's 0 1290555/16777216 = 0.076923072338104248 1 1290555/16777216 = 0.076923072338104248 2 1290555/16777216 = 0.076923072338104248 3 322639/4194304 = 0.0769231319427490234 4 1290555/16777216 = 0.076923072338104248 5 1290555/16777216 = 0.076923072338104248 6 1290555/16777216 = 0.076923072338104248 7 1290555/16777216 = 0.076923072338104248 8 1290555/16777216 = 0.076923072338104248 9 1290555/16777216 = 0.076923072338104248 10 1290555/16777216 = 0.076923072338104248 11 1290555/16777216 = 0.076923072338104248 12 1290555/16777216 = 0.076923072338104248 After 7's 0 1290555/16777216 = 0.076923072338104248 1 1290555/16777216 = 0.076923072338104248 2 1290555/16777216 = 0.076923072338104248 3 20648881/268435456 = 0.0769230760633945465 4 10324443/134217728 = 0.0769230946898460388 5 20648881/268435456 = 0.0769230760633945465 6 1290555/16777216 = 0.076923072338104248 7 1290555/16777216 = 0.076923072338104248 8 1290555/16777216 = 0.076923072338104248 9 1290555/16777216 = 0.076923072338104248 10 5162221/67108864 = 0.0769230872392654418 11 5162221/67108864 = 0.0769230872392654418 12 1290555/16777216 = 0.076923072338104248 After 8's 0 82595527/1073741824 = 0.0769230788573622703 1 82595527/1073741824 = 0.0769230788573622703 2 41297763/536870912 = 0.0769230779260396957 3 330382089/4294967296 = 0.0769230744335800409 4 165191045/2147483648 = 0.0769230746664106845 5 330382097/4294967296 = 0.0769230762962251901 6 165191051/2147483648 = 0.0769230774603784084 7 82595529/1073741824 = 0.0769230807200074195 8 165191051/2147483648 = 0.0769230774603784084 9 330382097/4294967296 = 0.0769230762962251901 10 165191045/2147483648 = 0.0769230746664106845 11 330382089/4294967296 = 0.0769230744335800409 12 41297763/536870912 = 0.0769230779260396957 After 9's 0 5286113585/68719476736 = 0.0769230767764383926 1 82595525/1073741824 = 0.0769230769947171211 2 5286113581/68719476736 = 0.0769230767182307317 3 5286113605/68719476736 = 0.0769230770674766972 4 2643056797/34359738368 = 0.0769230769074056297 5 5286113595/68719476736 = 0.0769230769219575449 6 5286113595/68719476736 = 0.0769230769219575449 7 2643056797/34359738368 = 0.0769230769074056297 8 5286113605/68719476736 = 0.0769230770674766972 9 5286113581/68719476736 = 0.0769230767182307317 10 82595525/1073741824 = 0.0769230769947171211 11 5286113585/68719476736 = 0.0769230767764383926 12 330382101/4294967296 = 0.0769230772275477647 After 10's 0 84577817515/1099511627776 = 0.0769230769174100714 1 84577817525/1099511627776 = 0.0769230769265050184 2 84577817531/1099511627776 = 0.0769230769319619867 3 21144454379/274877906944 = 0.0769230769183195661 4 84577817515/1099511627776 = 0.0769230769174100714 5 84577817519/1099511627776 = 0.0769230769210480502 6 42288908767/549755813888 = 0.0769230769346904708 7 84577817519/1099511627776 = 0.0769230769210480502 8 84577817515/1099511627776 = 0.0769230769174100714 9 21144454379/274877906944 = 0.0769230769183195661 10 84577817531/1099511627776 = 0.0769230769319619867 11 84577817525/1099511627776 = 0.0769230769265050184 12 84577817515/1099511627776 = 0.0769230769174100714 After 11's 0 338311270095/4398046511104 = 0.0769230769253681501 1 1353245080295/17592186044416 = 0.0769230769205364595 2 676622540193/8796093022208 = 0.0769230769257092106 3 1353245080295/17592186044416 = 0.0769230769205364595 4 338311270095/4398046511104 = 0.0769230769253681501 5 676622540153/8796093022208 = 0.0769230769211617371 6 1353245080365/17592186044416 = 0.0769230769245154988 7 338311270081/4398046511104 = 0.0769230769221849186 8 1353245080345/17592186044416 = 0.0769230769233786304 9 1353245080345/17592186044416 = 0.0769230769233786304 10 338311270081/4398046511104 = 0.0769230769221849186 11 1353245080365/17592186044416 = 0.0769230769245154988 12 676622540153/8796093022208 = 0.0769230769211617371 After 12's 0 5412980321359/70368744177664 = 0.0769230769230802025 1 21651921285435/281474976710656 = 0.0769230769230766497 2 21651921285435/281474976710656 = 0.0769230769230766497 3 21651921285435/281474976710656 = 0.0769230769230766497 4 21651921285435/281474976710656 = 0.0769230769230766497 5 21651921285435/281474976710656 = 0.0769230769230766497 6 21651921285435/281474976710656 = 0.0769230769230766497 7 21651921285435/281474976710656 = 0.0769230769230766497 8 21651921285435/281474976710656 = 0.0769230769230766497 9 21651921285435/281474976710656 = 0.0769230769230766497 10 21651921285435/281474976710656 = 0.0769230769230766497 11 21651921285435/281474976710656 = 0.0769230769230766497 12 21651921285435/281474976710656 = 0.0769230769230766497

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Answer	K Sengupta	2009-01-06 13:41:17
A minor observation	Steve Herman	2008-01-20 16:36:34
re: Computer Solution	Charlie	2008-01-16 12:35:41
Computer Solution	Oren	2008-01-16 04:19:51
Method	Praneeth	2008-01-16 04:06:35

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