Request for Odds Tutorial

I would like to learn some of the math behind calculating odds. Can someone help me? For instance, what are the odds of the three card flop in Texas Hold Em for the following (not considering the cards dealt to the players)?

Pair
All 3 Cards being red (or all 3 being black)
All 3 cards being odd (ace low and no face cards) or all 3 being even
Flush
Straight
A-2-3 (suited or unsuited)
All 3 Face Cards
Straight Flush (including A-K-Q suited)
Royal Flush

I have calculated some of the odds using an Excel formula as follows: (Are these correct?)

Pair = (COMBIN(13,1)*COMBIN(4,2)*COMBIN(50,1)-(13*12))/COMBIN(52,3) = 0.169411765. Then (1-0.169411765)/0.169411765 = 4.902777778 or 4.9 to 1.
All Red (or all black) = COMBIN(26,3)/COMBIN(52,3) = 0.117647059. Then (1-0.117647059)/0.117647059 = 7.5 or 7.5 to 1.
All odd or all even (no face card) = COMBIN(20,3)/COMBIN(52,3) = 0.05158371. Then (1-0.05158371)/0.05158371 = 18.38596491 or 18.4 to 1.
Flush = COMBIN(13,3)*COMBIN(4,1)/COMBIN(52,3) = 0.051764706. Then (1-0.051764706)/0.051764706 = 18.31818182 or 18.3 to 1.
Straight = COMBIN(12,1)*COMBIN(4,1)^3/COMBIN(52,3) = 0.033451131. Then (1-0.033451131)/0.033451131 = 27.77604167 or 27.8 to 1.
3 of a Kind = COMBIN(13,1)*COMBIN(4,3)/COMBIN(52,3) = 0.002352941. Then (1-0.002352941)/0.002352941 = 424 or 424 to 1.
Straight Flush (incl. A-K-Q suited) = COMBIN(12,1)*COMBIN(4,1)/COMBIN(52,3) = 0.002171946. Then (1-0.002171946)/0.002171946 = 459.416667 or 459.4 to 1.

How would I calculate an A-2-3 of any suit including a flush? How about all three cards being face cards? How about a three card royal flush on the flop?

Although I think I have the calculations above correct, how do each of these change if I use multiple decks? I cannot figure out the flush, straight, 3 of a kind, straight flush, or the others if I use two decks, three decks, eight decks, etc.

Can someone provide me a tutorial on these and correct my calculations using an Excel formula if I am wrong. Again, I am just wanting to learn the math behind calculating the odds.

Thank you for your help.

The Wizard has extensive resources on Texas Holdem probabilities at the Wizard of Odds site. The same website has an introduction to the Combinatorial and Factorial functions.

Thank you for your response. However, the main issue that I have that is not covered in his site is how the odds change if multiple decks of cards are used. (I would not expect that since Texas Holdem uses only one deck of cards.) Also, the sites that cover Texas Holdem really only address the odds of hitting your hand using the turn and the river card, again with one deck of cards. I am not looking for those types of odds. It is the multiple decks that is causing me the confusion.

Again, my question is, what are the odds of each of the hands using multiple decks of cards? 8 decks, 5 decks, 4 decks, etec. It is the changes from one deck that I cannot figure out. If there is a site that you can point to that would answer that, please let me know.

Quote: ceb

Thank you for your response. However, the main issue that I have that is not covered in his site is how the odds change if multiple decks of cards are used. (I would not expect that since Texas Holdem uses only one deck of cards.) Also, the sites that cover Texas Holdem really only address the odds of hitting your hand using the turn and the river card, again with one deck of cards. I am not looking for those types of odds. It is the multiple decks that is causing me the confusion.

Again, my question is, what are the odds of each of the hands using multiple decks of cards? 8 decks, 5 decks, 4 decks, etec. It is the changes from one deck that I cannot figure out. If there is a site that you can point to that would answer that, please let me know.

Scroll down this page for a table of multiple deck probabilities.

I saw that before, but the multiple deck probabilities are for a 5 card poker hand, not a 3 card poker hand and they do not show how it is calculated. I also saw further below that table that there are numbers from 2 cards to 10 cards, however, that again does not show the odds that I am looking for. For instance, I already know that there are only 4 ways to have a mini royal flush with 3 cards using one deck of cards. However, with two decks of cards there are 8 ways to make a mini royal flush using three cards. With 3 decks of cards there are 27 ways to make a three card mini royal flush, etc. I have only been able to do that because the calculations are easy. However, how does one make the change in the formula for 6 decks or 8 decks. Also, how does a person make the changes in the formula for a straight flush, straight, flush, 3 of a kind, etc. It is really the formula that I am looking to better understand how these odds are calculated. Thanks again for the help.

Quote: ceb

... with two decks of cards there are 8 ways to make a mini royal flush using three cards. ...

More than that, I think, since the six "royal" cards in each suit can be gathered in several ways to form a mini royal flush.

Sorry. You're right. There are 8 ways per suit or 32 ways for two decks of cards. The 3-decks of cards has four times as much as well or 108 ways. Thanks for noticing my error.

As promised here is what I came up with. Hope there are no brainos and/or typos. The denominator is combin(52*d,3). Any questions, just ask. Also note that there can be overlaps. Pairs can end up being flushes (2 queen of spades and a king of spades for example). Obviously the straight flushes are included in the straights.
pairs: there are 13 ranks for the pair There are 4 of each rank per deck. Pick any 2 of those cards combin(4*d,2). There are 12 ranks for the other card and 4 of each rank per deck. 13*combin(4*d,2)*12*4*d

Red: There are 26 red cards per deck. Pick and 3 of those cards. combin(26*d,3).

Odd: There are 20 odd cards per deck. Pick and 3 of those cards. combin(20*d,3).

Flush: There are 4 suits. There are 13 of each suit per deck. Pick any 3 of those cards. 4*combin(13*d,3)

Straights: There are 12 ranks that can be the high card in the straight (3-10,j,q,k,a) and 4 suits per deck. 12*4*d. There are 4 suits per deck for each of the middle and low cards. 12*4*d*4*d*4*d

Straightflush: There are 12 ranks that can be the high card in the straightflush (3-10,j,q,k,a) and 4 suits per deck. 12*4*d. There is 1 card per deck for each of the middle and low cards. 12*4*d*d*d

3 of a kind: There are 13 ranks to choose from. There are 4 of each rank per deck. Pick any 3 of those cards. 13*combin(4*d,3)

Mini Royal: There are 4 suits to choose from. There is one ace, king, and queen of that suit per deck. 4*d*d*d

Wheel: There are 4 aces, twos and threes per deck. 4*d*4*d*4*d

3 Facecards: There are 12 face cards per deck. Pick any 3 of those cards. combin(12*d,3)

Thank you for your help.