I'd count the offsuit high cards, such as a hand like AQT4sK, so that your 27 should be a 33.
----- h_dunbar@hotmail.com [vpFREE] <vpFREE@yahoogroups.com> wrote:
>I agree with Albert that the overall cost is extremely small when individual cost is small and the play is rare.
We're talking about the specific 3-card RF of Ace, Ten, with a J, Q or K.
When we have that hand in JOB, holding the 4th flush card costs 1% of our bet. (5 cents on a $5 bet.) How often do we get the hand?
I think we get that hand about 0.1% of the time. (my calc is shown below)
So, by keeping the 4th flush card, we will lower our EV by 1% of 0.1% of our total bets. For every $1 million we bet, we give up $10 in EV by holding the 4th flush card.
How did I get the 0.1% figure? I'm going to take the time to write it out because (1) if I've done it correctly, it may help someone do similar calc's, and (2) if I'm wrong, I want to know where!
So, here goes: How many ways are there to get a hand with A, T, another RF card, and a 4th suited card?
Say I want to know the chance of getting 5 cards in this order:
Ace, Ten, otherRFcard, other suited card, other non-suited card
Card Ways Why
Ace 4 4 suits to choose from
Ten 1 has to be same suit
OtherRF card 3 can be J, Q, or K of same suit
4th flush card 8 can only be 2-9, same suit
5th card 27 can be 2-10 of any different suit
To get the total number of ways to deal out a hand in that order, you multiply the "Ways". 4 * 1 * 3 * 8 * 27 = 2592
But we don't care what order the cards are in, so we have to figure out how many ways the 5 cards can be ordered.
There are 5 ways to pick which card will be 1st, 4 ways to pick the 2nd card, 3 ways to pick the 3rd card, 2 ways to pick the 4th card, and just 1 way to pick the 5th card. Multiplying 5*4*3*2*1 makes 120 ways to order the five different cards.
So, the total number of ways to be dealt a hand with A,T,(J,Q,or K), small suited, and small non-suited is 2592 * 120 = 311,040.
That seems like a lot, but it's a small fraction of the 52 * 51 * 50 * 49 * 48 ways to deal out ANY five cards. That number is 311,875,200.
Therefore, the chance of being dealt a 3-card RF with A,T with a 4th small (2-9) flush card and a small (2-10) non-suited card is 311,040 / 311,875,200.= 0.1%
And the total cost of holding the 4th flush card on that hand is 1% of 0.1% = 0.001% of your total bet.
I'm not endorsing holding the 4th flush card as a general play in JOB. As "5-card" mentioned, the cost is much bigger on other 3-card RF's. The Ace/Ten 3-card RF's are only a small fraction of the total times you might have a 3-card RF with a 4th flush card.
--Dunbar
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Posted by: 007 <007@embarqmail.com>
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