From Laquita G.
Hello Stickman,
I very much enjoy your articles. I find the information you present interesting and easy to understand.
On to my question for you. Every time I am dealt a hand containing a three of a kind, I know I am a winner and I know the winning amount is more than just my initial bet. What I would like to know is what are my odds of improving that hand, you know, how many times will I get a full house and how many times will I get a four of a kind and how many times will I have to “settle” for just the 3-for-1 payout from the initial hand. This may be a petty question, but inquiring minds want to know.
Laquita G, Georgia
Hi Laquita,
Thank you for the kind comments. On certain hands, I like to know what the odds are of completing them, rather than improving them. I keep track of the number of times I am dealt four cards of straight (separate counts for inside and open-ended), four cards of a flush, four cards of a straight flush (inside and open-ended), and four cards of a royal flush. I also keep track of how often they are completed.
Depending on the type of hand dealt, the number of cards that will complete the hand are:
• Inside straight – four cards
• Open straight – eight cards
• Flush – nine cards
• Inside straight flush – one card
• Open straight flush – two cards
• Royal Flush – one card
From here, the odds are straightforward to determine. Since five cards have been dealt, there are 47 cards left in the deck (I do not play wild card video poker as a rule). The odds of completing these hands are:
• Inside straight – four in 47 or about one in 12 (11-3/4)
• Open straight – eight in 47 or about one in six (5-7/8)
• Flush – nine in 47 or about one in five (5-2/9)
• Inside straight flush – one in 47
• Open straight flush – two in 47 or one in 23-1/2
• Royal Flush – one in 47
Now let us tackle your question.
Since three cards are being held there are two open positions for the final hand. 47 possible cards for the first open slot and 46 possible cards for the last open spot. The formula to calculate the number of possible hands is:
• Number of cards available for the first open slot times the number of cards available for the second open slot divided by the number of slots (47 * 46 / 2).
• 47 * 46 = 2,162.
• 2,162 / 2 = 1,081.
There are 1,081 possible hands when holding three cards.
Of these 1,081 hands:
• 66 will be a full house
• 46 will be four-of-a-kind
• 969 will be the original three-of-a-kind
The odds of improving the hand are 66+46 or 112 in 1,081. Putting this in the simplest terms, about one in 10 – actually a little better than one in ten – one in 9.65.
This number is “on average.” You will definitely not get an improved hand after each 10 hands. The number of hands with no improvement between hands that are improved can vary significantly. There may be 10 hands between improved hands, 50 hands, 5 hands, 0 hands (back-to-back improved hands), or 100 hands. In the long run, however, the average will be very close to one in 9.65 hands.
Thank you for this question, Laquita. I have never even thought about looking at the odds of improving hands that are dealt winners, but it is a great idea and it was a fun exercise.
As always, may all your wins be swift and large, and your losses be slow and small.
Jerry “Stickman”
Jerry “Stickman” is an expert in craps, blackjack and video poker and advantage slot machine play. He is a regular contributor to top gaming magazines and 888casino.com. He authored the video poker section of "Everything Casino Poker: Get the Edge at Video Poker, Texas Hold'em, Omaha Hi-Lo, and Pai Gow Poker!" You can contact Jerry “Stickman” at stickmanjerryg@gmail.com.
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