Averzhu's Blog

The number of possible royal flushes for the player's hand = 4

times

the number of possible dealer hands which qualify but do not tie the player's hand = 837987

times

the number of cards in the dealer's hand which could be the one that gets exposed = 5



4 * 837987 * 5 = 16759740

Thank you very much, BJ. Can you give me the Mathematical formulas of "the number of possible dealer hands which qualify but do not tie the player's hand" Combinations? Because I'm a beginner. Thanks.

Sorry, should thank JB.

The combinations of Five Card Stud Poker(MS Excel Function):



royal flush: combin(1,1)*combin(4,1)=4

straight flush: combin(10,1)*combin(4,1)-4=36

four of a kind: combin(13,1)*combin(4,4)*combin(52-4,1)=624

full house: combin(13,1)*combin(4,3)*combin(13-1,1)*combin(4,2)=3744

flush: combin(13,5)*combin(4,1)-4-36=5108

straight: combin(10,1)*power(combin(4,1),5)-4-36=10200

three of a kind: combin(13,1)*combin(4,3)*combin(13-1,2)*power(combin(4,1),2)=54912

two pair: combin(13,2)*power(combin(4,2),2)*combin(13-2,1)*combin(4,1)=123552

high card: (combin(13,5)-10)*(power(combin(4,1),5)-4)=1302540

Total: combin(52,5)=2598960



Five Card Draw -- High Card Hands:

Ace high: (combin(12,4)-2)*(power(combin(4,1),5)-4)=502860

King high: (combin(11,4)-1)*(power(combin(4,1),5)-4)=335580

Queen high: (combin(10,4)-1)*(power(combin(4,1),5)-4)=213180

Jack high: (combin(9,4)-1)*(power(combin(4,1),5)-4)=127500

10 high: (combin(8,4)-1)*(power(combin(4,1),5)-4)=70380

9 high: (combin(7,4)-1)*(power(combin(4,1),5)-4)=34680

8 high: (combin(6,4)-1)*(power(combin(4,1),5)-4)=14280

7 high: (combin(5,4)-1)*(power(combin(4,1),5)-4)=4080

Total: 1302540



Ace&King High: (combin(11,3)-1)*(power(combin(4,1),5)-4)=167280



The Link:

http://wizardofodds.com/games/poker/

one pair: combin(13,1)*combin(4,2)*combin(13-1,3)*power(combin(4,1),3)=1098240

I don't use formulas to calculate that, as you have to take into account the cards in the player's hand which cannot appear in the dealer's hand. It is best to write a computer program to iterate through the possible dealer hands.

Thanks.I will try.

I have written a Visual Basic program to calculate the "Player wins with royal flush" Combinations.

The answer is :

837987 PlayerRoyalFlushWin

3 Push

695949 NonQuality

1533939 Total

------------------------------------------------------------------------------

Private Sub Command1_Click()



Dim MyHands As String



MyHands = PlayerDealerCompare("01S10S11S12S13S")



Text1.Text = MyHands



End Sub

----------------------------------------------------------------------------

Public Function PlayerDealerCompare(PlayerHand As String)



Const CardDraw As Integer = 5

Dim i, j, k, l, m, CardNo, SearchCard, PlayerStatus As Integer, DealerStatus As Integer

Dim CardValue As String, Flowers(1 To 4) As String * 1

Dim DealerDecks(1 To 47) As String, DealerOutput As String

Dim DealerHand As String

Dim PlayerRFWin, PlayerRFPush, PlayerRFNonQuality

Dim DealerAK As Boolean



Flowers(1) = "S"

Flowers(2) = "H"

Flowers(3) = "C"

Flowers(4) = "D"



For i = 1 To 13



If Len(Trim(Str(i))) = 1 Then

CardValue = "0" + Trim(Str(i))

Else

CardValue = Trim(Str(i))

End If



For j = 1 To 4

SearchCard = InStr(1, PlayerHand, CardValue + Flowers(j), 1)

If SearchCard = 0 Then

CardNo = 1 + CardNo

DealerDecks(CardNo) = CardValue + Flowers(j)

End If

Next



Next



For i = 1 To CardNo - CardDraw + 1

For j = i + 1 To CardNo - CardDraw + 2

For k = j + 1 To CardNo - CardDraw + 3

For l = k + 1 To CardNo - CardDraw + 4

For m = l + 1 To CardNo - CardDraw + 5



DealerHand = DealerDecks(i) + DealerDecks(j) + DealerDecks(k) + DealerDecks(l) + DealerDecks(m)



PlayerStatus = CardStatus(PlayerHand)

DealerStatus = CardStatus(DealerHand)



Select Case PlayerStatus

Case 0



Case 1



Case 2



Case 3



Case 4



Case 5



Case 6



Case 7



Case 8



Case 9

If PlayerStatus > DealerStatus And DealerStatus <> 0 Then



PlayerRFWin = PlayerRFWin + 1



ElseIf PlayerStatus = DealerStatus Then



PlayerRFPush = PlayerRFPush + 1



ElseIf DealerStatus = 0 Then



DealerAK = AKStatus(DealerHand)



If DealerAK = True Then

PlayerRFWin = PlayerRFWin + 1

Else

PlayerRFNonQuality = PlayerRFNonQuality + 1

End If



End If

End Select

Next

Next

Next

Next

Next



DealerOutput = Str(PlayerRFWin) + " PlayerRoyalFlushWin" + vbCrLf _

+ Str(PlayerRFPush) + " Push" + vbCrLf _

+ Str(PlayerRFNonQuality) + " NonQuality" + vbCrLf _

+ Str(PlayerRFWin + PlayerRFPush + PlayerRFNonQuality) + " Total"



PlayerDealerCompare = DealerOutput

End Function

----------------------------------------------------------------------------------------------------------------------

Public Function CardStatus(CardHand As String)



Dim CardNo(1 To 5) As String, CardFlower(1 To 5) As String

Dim ii, jj, SameCard, Pair, Straight, HowManySameCard, SameFlower, ThisNo, NextNo

Dim Status As String, Flower As String, ComCard As String

Dim Royal As Boolean



SameCard = 0

Pair = 0

Straight = 0

SameFlower = 0

HowManySameCard = 0

Royal = False



jj = 1

For ii = 1 To 13 Step 3

CardNo(jj) = Mid(CardHand, ii, 2)

CardFlower(jj) = Mid(CardHand, ii + 2, 1)

Status = Status + CardNo(jj)

Flower = Flower + CardFlower(jj)

jj = jj + 1

Next



For ii = 1 To 4 'One pair or two pairs(1or2)

If ComCard <> CardNo(ii) Then

For jj = ii + 1 To 5

If StrComp(CardNo(ii), CardNo(jj)) = 0 Then

SameCard = SameCard + 1

ComCard = CardNo(ii)

End If

Next



If SameCard = 1 Then Pair = Pair + 1

End If



SameCard = 0

Next



If Pair = 1 Then HowManySameCard = 1 'One pair is 1

If Pair = 2 Then HowManySameCard = 2 'Two pairs is 2







For ii = 1 To 4

ThisNo = CInt(CardNo(ii))

NextNo = CInt(CardNo(ii + 1))



If ThisNo = NextNo - 1 Then Straight = Straight + 1 'Straight





If StrComp(CardFlower(1), CardFlower(ii + 1)) = 0 Then 'Flush

SameFlower = SameFlower + 1

End If

Next



ComCard = ""

For ii = 1 To 4 'three of a kind is 3 or four of a kind is 7

If ComCard <> CardNo(ii) Then

For jj = ii + 1 To 5

If StrComp(CardNo(ii), CardNo(jj)) = 0 Then

SameCard = SameCard + 1

ComCard = CardNo(ii)

End If

Next

End If



Select Case SameCard

Case 2 'three of a kind is 3

HowManySameCard = 3

Case 3 'four of a kind is 7

HowManySameCard = 7

End Select





SameCard = 0

Next



If CardNo(1) = "01" And CardNo(2) = "10" And CardNo(3) = "11" And _

CardNo(4) = "12" And CardNo(5) = "13" Then Royal = True



If Pair = 1 And HowManySameCard = 3 Then HowManySameCard = 6 'full house is 6

If SameFlower = 4 Then HowManySameCard = 5 'flush is 5

If Straight = 4 Or Royal Then HowManySameCard = 4 'Straight is 4

If Straight = 4 And SameFlower = 4 Then HowManySameCard = 8 'straight flush is 8



If CardNo(1) = "01" And CardNo(2) = "10" And CardNo(3) = "11" And _

CardNo(4) = "12" And CardNo(5) = "13" And SameFlower = 4 Then _

HowManySameCard = 9 'royal flush



CardStatus = HowManySameCard



End Function

-------------------------------------------------------------------------------------------------------------

Public Function AKStatus(CardHand As String)



Dim HandAK As Boolean

HandAK = False



If InStr(1, CardHand, "01", 1) > 0 And InStr(1, CardHand, "13", 1) > 0 Then

HandAK = True

Else

HandAK = False

End If



AKStatus = HandAK

End Function

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