Card Game

[jasonHoldEm]

Awesome, nothing like some hardcore math and data to get the zoo running in top form.

My work here is done, thanks to all for the help on problem 2. Anyone who thinks the zoo is dead just needs to see a thread like this occasionally to see we still got it.

J

[GrannyMae]

1 1 0.019230769 1 0.019230769
2 2 0.038461538 0.980769231 0.037721893
3 3 0.057692308 0.943047337 0.054406577
4 4 0.076923077 0.88864076 0.068356982
5 5 0.096153846 0.820283779 0.07887344
6 6 0.115384615 0.741410338 0.085547347
7 7 0.134615385 0.655862992 0.088289249
8 8 0.153846154 0.567573743 0.087319037
9 9 0.173076923 0.480254705 0.083121007
10 10 0.192307692 0.397133699 0.076371865
11 11 0.211538462 0.320761834 0.067853465
12 12 0.230769231 0.252908369 0.05836347
13 13 0.25 0.194544899 0.048636225
14 14 0.269230769 0.145908674 0.039283105
15 15 0.288461538 0.10662557 0.030757376
16 16 0.307692308 0.075868194 0.02334406
17 17 0.326923077 0.052524134 0.017171352
18 18 0.346153846 0.035352783 0.012237502
19 19 0.365384615 0.023115281 0.008445968
20 20 0.384615385 0.014669313 0.005642043
21 21 0.403846154 0.009027269 0.003645628
22 22 0.423076923 0.005381641 0.002276848
23 23 0.442307692 0.003104793 0.001373274
24 24 0.461538462 0.001731519 0.000799163
25 25 0.480769231 0.000932357 0.000448248
26 26 0.5 0.000484108 0.000242054
27 27 0.519230769 0.000242054 0.000125682
28 28 0.538461538 0.000116372 6.26619E-05
29 29 0.557692308 5.37102E-05 2.99538E-05
30 30 0.576923077 2.37564E-05 1.37056E-05
31 31 0.596153846 1.00508E-05 5.99183E-06
32 32 0.615384615 4.05898E-06 2.49783E-06
33 33 0.634615385 1.56115E-06 9.90727E-07
34 34 0.653846154 5.70419E-07 3.72966E-07
35 35 0.673076923 1.97453E-07 1.32901E-07
36 36 0.692307692 6.45518E-08 4.46897E-08
37 37 0.711538462 1.98621E-08 1.41326E-08
38 38 0.730769231 5.72945E-09 4.18691E-09
39 39 0.75 1.54254E-09 1.15691E-09
40 40 0.769230769 3.85636E-10 2.96643E-10
41 41 0.788461538 8.89929E-11 7.01675E-11
42 42 0.807692308 1.88254E-11 1.52052E-11
43 43 0.826923077 3.62028E-12 2.99369E-12
44 44 0.846153846 6.26586E-13 5.30188E-13
45 45 0.865384615 9.63979E-14 8.34212E-14
46 46 0.884615385 1.29766E-14 1.14793E-14
47 47 0.903846154 1.4973E-15 1.35333E-15
48 48 0.923076923 1.43972E-16 1.32897E-16
49 49 0.942307692 1.10747E-17 1.04358E-17
50 50 0.961538462 6.38927E-19 6.14353E-19
51 51 0.980769231 2.45741E-20 2.41015E-20
52 52 1 4.72579E-22 4.72579E-22
1

[GrannyMae]

is that punk's castle on the left? also, i would absolutely love to own that car. seriously.
i'll bet you could recoup anything you might pay for it just by salvaging those intact windows and reselling them as spare parts.
(unless that is saran wrap)

[Lori]

That's Eugene's car.

Lori

[34TheTruth34]

[ QUOTE ]
all have an equal chance if the deck is shuffled properly, so i would guess the last position is best.


[/ QUOTE ]

wha???

[Cazz]

I got the same answer.

Simplied formula

insert formula into D2
=(ROW(D2)-1)/52*(1-SUM($D$11))
copy and paste d2-d53
You can put 1-52 in C2 thru C53 and replace the stuff
inside the first () w/ C2.

[Homer]

I sort of skimmed the thread and didn't see any simple math provided for Problem 1...

Probability of winning

Contestant #1 = 1/52
Contestant #2 = (51/52)*(1/51) = 1/52
Contestant #3 = (51/52)*(50/51)*(1/50) = 1/52

...and so on.

-- Homer

[SamJack]

Jek and whoever agreed with him has the right answer.

Since homer provided the formula for #1.
Here it is for #2.

For N, the probability you will be the winner is:

(Probability that you will have ace spade out of the N cards you are allowed to pick) * (probability that the people who went before you did not win)

N/52 * (1 - Sum(probofwinning(1) to probofwinning(n-1)))

Sorry I can't type the mathamatical notation, but you get the idea.

SamJack

[pzhon]

Problem 2: A simpler way to express the probability of winning in position N is

(51/52)(50/52)(49/52)...((52-N+1)/52) * (N/52).

The probability that the Nth person wins divided by the probability that the N+1st person wins is

(52/(52-N)) (N/N+1).

This should switch from greater than 1 to less than 1 at the maximum. The maximum should be close to where this ratio equals 1 for some real N.

(52/(52-N)) (N/N+1) = 1
...
(N + 1/2)^2 = 52.25

N = Sqrt(52.25)-.5 = 6.728.

For smaller values including N=6, the Nth person wins less frequently than the N+1st person wins. So, person 7 does better than person 6.

For larger values including N=7, the Nth person wins more than the N+1st person wins. So, person 7 does better than person 8.

With c cards in the deck, the probability of winning is greatest for the person in the position closest to sqrt(c+.25). For c=20, sqrt(c+.25) = 4.5, and positions 4 and 5 have equal chances.