What is the wrong in my code in mathematica - wolfram-mathematica

Question

Assume that you play a variant of a poker game with five cards that is using a subset of one ordinary deck of cards. The subset considered is cards valued 7, 8, 9,10,J,Q,K,A from the one decks, in total 32 cards. Allow A-7-8-9-10 as a low straight or straight flush.
Calculate the exact probability of the following hands using the census method.
one pair
two pairs
three of a kind
straight
full house
flush
four of a kind
straight flush
nothing
Compare and discuss the probabilities of the above variant poker game to the probabilities for a normal poker game with a standard 52 card deck.
Note: If a given hand contains different valued hands only one (the highest rank) is considered
my solution was:
Any hand
In[172]:=
In[263]:= deck = Range[7, 13]
Out[263]= {7, 8, 9, 10, 11, 12, 13}
Subsets[{7, 8, 9, 10, A, J, Q, K}, {5}]
Out[41]= {{7, 8, 9, 10, A}, {7, 8, 9, 10, J}, {7, 8, 9, 10, Q}, {7, 8, 9, 10, K}, {7,
8, 9, A, J}, {7, 8, 9, A, Q}, {7, 8, 9, A, K}, {7, 8, 9, J, Q}, {7, 8, 9, J,
K}, {7, 8, 9, Q, K}, {7, 8, 10, A, J}, {7, 8, 10, A, Q}, {7, 8, 10, A,
K}, {7, 8, 10, J, Q}, {7, 8, 10, J, K}, {7, 8, 10, Q, K}, {7, 8, A, J,
Q}, {7, 8, A, J, K}, {7, 8, A, Q, K}, {7, 8, J, Q, K}, {7, 9, 10, A, J}, {7,
9, 10, A, Q}, {7, 9, 10, A, K}, {7, 9, 10, J, Q}, {7, 9, 10, J, K}, {7, 9,
10, Q, K}, {7, 9, A, J, Q}, {7, 9, A, J, K}, {7, 9, A, Q, K}, {7, 9, J, Q,
K}, {7, 10, A, J, Q}, {7, 10, A, J, K}, {7, 10, A, Q, K}, {7, 10, J, Q,
K}, {7, A, J, Q, K}, {8, 9, 10, A, J}, {8, 9, 10, A, Q}, {8, 9, 10, A,
K}, {8, 9, 10, J, Q}, {8, 9, 10, J, K}, {8, 9, 10, Q, K}, {8, 9, A, J,
Q}, {8, 9, A, J, K}, {8, 9, A, Q, K}, {8, 9, J, Q, K}, {8, 10, A, J, Q}, {8,
10, A, J, K}, {8, 10, A, Q, K}, {8, 10, J, Q, K}, {8, A, J, Q, K}, {9, 10,
A, J, Q}, {9, 10, A, J, K}, {9, 10, A, Q, K}, {9, 10, J, Q, K}, {9, A, J, Q,
K}, {10, A, J, Q, K}}
Length[Subsets[{7, 8, 9, 10, A, J, Q, K}, {5}]]
In[259]:= Subsets[{A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K}, {5}]
In[260]:= Length[Subsets[{A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K}, {5}]]
Out[260]= 1287
One pair:
32kort:
In[135]:= deck = Sort[Join[Range[7, 14], Range[7, 14], Range[7, 14], Range[7, 14]]];
hands = Subsets[deck, {5}];
onePairQ32[{___, x_, x_, ___}] := True;
onePairQ32[{___, x_, x_, x_, ___}] := False;
onePairQ32[{___, x_, x_, ___, y_, y_, ___} /; x != y] := False;
onePairQ32[{___}] := False;
Count[hands, _?onePairQ32]
Out[141]= 107520
52kort:
In[128]:=
deck = Sort[Join[Range[1, 13], Range[1, 13], Range[1, 13], Range[1, 13]]];
hands = Subsets[deck, {5}];
onePairQ52[{___, x_, x_, ___}] := True;
onePairQ52[{___, x_, x_, x_, ___}] := False;
onePairQ52[{___, x_, x_, ___, y_, y_, ___} /; x != y] := False;
onePairQ52[{___}] := False;
Count[hands, _?onePairQ52]
Out[134]= 1098240
Two pairs:
32kort:
In[8]:= deck = Sort[Join[Range[7, 14], Range[7, 14], Range[7, 14], Range[7, 14]]];
hands = Subsets[deck, {5}];
TwoPairQ32[{___, x_, x_, ___}] := False;
TwoPairQ32[{___, x_, x_, x_, ___}] := False;
TwoPairQ32[{___, x_, x_, ___, y_, y_, ___} /; x != y] := True;
TwoPairQ32[{___}] := False;
Count[hands, _?TwoPairQ32]
Out[14]= 24192
52kort:
In[15]:= deck = Sort[Join[Range[1, 13], Range[1, 13], Range[1, 13], Range[1, 13]]];
hands = Subsets[deck, {5}];
TwoPairQ52[{___, x_, x_, ___}] := False;
TwoPairQ52[{___, x_, x_, x_, ___}] := False;
TwoPairQ52[{___, x_, x_, ___, y_, y_, ___} /; x != y] := True;
TwoPairQ52[{___}] := False;
Count[hands, _?TwoPairQ52]
Out[21]= 123552
Three of a kind:
32kort:
In[22]:= deck = Sort[Join[Range[7, 14], Range[7, 14], Range[7, 14], Range[7, 14]]];
hands = Subsets[deck, {5}];
ThreeOfKindQ32[{___, x_, x_, ___}] := False;
ThreeOfKindQ32[{___, x_, x_, x_, ___}] := True;
ThreeOfKindQ32[{___, x_, x_, ___, y_, y_, ___} /; x != y] := False;
ThreeOfKindQ32[{___}] := False;
Count[hands, _?ThreeOfKindQ32]
Out[28]= 12320
52kort:
In[142]:= deck = Sort[Join[Range[1, 13], Range[1, 13], Range[1, 13], Range[1, 13]]];
hands = Subsets[deck, {5}];
ThreeOfKindQ52[{___, x_, x_, ___}] := False;
ThreeOfKindQ52[{___, x_, x_, x_, ___}] := True;
ThreeOfKindQ52[{___, x_, x_, ___, y_, y_, ___} /; x != y] := False;
ThreeOfKindQ52[{___}] := False;
Count[hands, _?ThreeOfKindQ52]
Out[148]= 59280
pairQ[{x_, x_, y_, y_} /; x != y] := False; (* two pairs *)
pairQ[{___, x_, x_, x_, ___}] := False; (* three of a kind *)
pairQ[{___, x_, x_, ___}] := True; (* a pair *)
pairQ[{___}] := False (* else *)
Four of a kind:
32kort:
In[36]:= deck = Sort[Join[Range[7, 14], Range[7, 14], Range[7, 14], Range[7, 14]]];
hands = Subsets[deck, {5}];
FourOfKindQ32[{___, x_, x_, ___}] := False;
FourOfKindQ32[{___, x_, x_, x_, ___}] := False;
FourOfKindQ32[{___, x_, x_, ___, y_, y_, ___} /; x != y] := False;
FourOfKindQ32[{___}] := False;
FourOfKindQ32[{___, x_, x_, x_, x_, ___}] := True;
Count[hands, _?FourOfKindQ32]
Out[43]= 224
52kort:
In[44]:= deck = Sort[Join[Range[1, 13], Range[1, 13], Range[1, 13], Range[1, 13]]];
hands = Subsets[deck, {5}];
FourOfKindQ52[{___, x_, x_, ___}] := False;
FourOfKindQ52[{___, x_, x_, x_, ___}] := False;
FourOfKindQ52[{___, x_, x_, ___, y_, y_, ___} /; x != y] := False;
FourOfKindQ52[{___}] := False;
FourOfKindQ52[{___, x_, x_, x_, x_, ___}] := True;
Count[hands, _?FourOfKindQ52]
Out[51]= 624
Full house:
32kort:
In[98]:= deck = Sort[Join[Range[7, 14], Range[7, 14], Range[7, 14], Range[7, 14]]];
hands = Subsets[deck, {5}];
FullhouseQ32[{___, x_, x_, ___}] := False;
FullhouseQ32[{___, x_, x_, x_, ___}] := False;
FullhouseQ32[{___, x_, x_, ___, y_, y_, ___} /; x != y] := False;
FullhouseQ32[{___}] := False;
FullhouseQ32[{x_, x_, x_, y_, y_} /; x != y] := True;
Count[hands, _?FullhouseQ32]
Out[105]= 672
52kort:
In[157]:= deck = Sort[Join[Range[1, 13], Range[1, 13], Range[1, 13], Range[1, 13]]];
hands = Subsets[deck, {5}];
FullhouseQ52[{___, x_, x_, ___}] := False;
FullhouseQ52[{___, x_, x_, x_, ___}] := False;
FullhouseQ52[{___, x_, x_, ___, y_, y_, ___} /; x != y] := False;
FullhouseQ52[{___}] := False;
FullhouseQ52[{x_, x_, y_, y_, y_} /; x != y] := True;
Count[hands, _?FullhouseQ52]
Out[164]= 1872
Nothing:
32kort:
In[75]:= deck = Sort[Join[Range[7, 14], Range[7, 14], Range[7, 14], Range[7, 14]]];
hands = Subsets[deck, {5}];
NothingQ32[{___, x_, x_, ___}] := False;
NothingQ32[{___, x_, x_, x_, ___}] := False;
NothingQ32[{___, x_, x_, ___, y_, y_, ___} /; x != y] := False;
NothingQ32[{___}] := True;
Count[hands, _?NothingQ32]
Out[81]= 57344
52kort:
In[165]:= deck = Sort[Join[Range[1, 13], Range[1, 13], Range[1, 13], Range[1, 13]]];
hands = Subsets[deck, {5}];
NothingQ52[{___, x_, x_, ___}] := False;
NothingQ52[{___, x_, x_, x_, ___}] := False;
NothingQ52[{___, x_, x_, ___, y_, y_, ___} /; x != y] := False;
NothingQ52[{___}] := True;
Count[hands, _?NothingQ52]
Out[171]= 1317888
I got wrong values but I don't know what my problem is

Answer 1

Your 3-of-a-kind hands includes 4-of-a-kind hands.
Examples:
Take[Union[Cases[hands,_?ThreeOfKindQ32]],20]
which displays
{{7, 7, 7, 7, 8}, {7, 7, 7, 7, 9}, {7, 7, 7, 7, 10}, {7, 7, 7, 7, 11},
{7, 7, 7, 7, 12}, {7, 7, 7, 7, 13}, {7, 7, 7, 7, 14}, {7, 7, 7, 8, 8},
{7, 7, 7, 8, 9}, {7, 7, 7, 8, 10}, {7, 7, 7, 8, 11}, {7, 7, 7, 8, 12},
{7, 7, 7, 8, 13}, {7, 7, 7, 8, 14}, {7, 7, 7, 9, 9}, {7, 7, 7, 9, 10},
{7, 7, 7, 9, 11}, {7, 7, 7, 9, 12}, {7, 7, 7, 9, 13}, {7, 7, 7, 9, 14}}
Next, either I've made a mistake or there is something wrong with your Full House patterns. When I scrape-n-paste your code for that it gives me zero for a count. I have not figured out what is going on with that. Can you start a fresh notebook and paste nothing but your posted code for that and evaluate it and see if you also get zero?
When I change that code to
FullhouseQ32[{___}] := False;
FullhouseQ32[{x_, x_, x_, y_, y_} /; x != y] := True;
FullhouseQ32[{x_, x_, y_, y_, y_} /; x != y] := True;
I get what seems to be correct result, but I can't be certain.