Some Important Probabilities!!

[Dazzler69]

I saw this and thought it would be very useful for beginners as some of the percentages are a lot higher than I thought they would be.

Below is a table which shows the probabilty that an overcard will appear on the flop and on the board in total when you have pocket pair

Pocket Pair............Overcard On Flop..........Overcard On Board

2-2.............................1.00................ ............1.00
3-3.............................0.999..........................0.999997
4-4.............................0.994 .........................0.99988
5-5.............................0.981 .........................0.9991
6-6.............................0.958............... ...........0.996
7-7.............................0.921............... ...........0.988
8-8.............................0.867............... ...........0.969
9-9.............................0.793............... ...........0.933
10-10..........................0.695................. .........0.869
J-J..............................0.570.............. ............0.763
Q-Q............................0.414................ ..........0.599
K-K.............................0.226............... ...........0.353
A-A.............................0.00................ ............0.00


Hope this is of some use to at least one person, if you want to know the maths that goes into to working out these proabilities, then you need to get out more, no I'm just joking just ask and from my limited knowledge of maths I will attempt to explain it because I do have the explanation its just a bit detailed to write unless someone specifically wants to know.


N.B - Apoligises for all the dots but that is the only way I could do it so that it all lined up properly as I couldn't put a table or anything in

[scrawnybob]

hi Dazzler

nice post - be great to have the explaination etc you mentioned - think it would be really useful with the data you've posted

cheers scrawnybob

[Dazzler69]

Here goes, I tried explaining this the other day and the person just got lost but I'll try.

I'll Q-Q as an example. If you have Q-Q before the flop then there are 50 cards that you cannot see. Therefore we are getting 3 cards from this 50 in order to make the flop. There C(50,3)=50!/3!47! = 19,600 possible flops.

What we are interested in is how many of these flops will contain an Ace or a King (as they are overcards to the pocket Queens). We figure this out by working how many flops do not contain an Ace or King and subtracting it from the total number of flops (19,600).

So they are 50 remaining cards after taking out the QQ in our hand, once we take out the Aces and Kings there 42 cards left giving us
C(42,3) = 42!/3!39! = 11,480 flops that do not contain an Ace or a King.

So we take the 11,480 flops that have no Ace or King and subtract it from the total number of flops (19,600)

19,600 - 11,480 = 8,120

So there are 8,120 flops that will contain an Ace or King, and the probabilty of and an overcard to the QQ appearing is 8,120/19,600 = 0.414 or 41% of the time.

To calculate the probabilty of an over card on the whole board similar calculations are performed. There are C(50,5) = 2,118,760 possible boards and there are C(42,5) = 850,668 boards without an Ace or a King. So the number of boards with an overcard is 1,268,092. Therefore the probabilty of an overcard appearing on the board is 0.599 or 60%.

N.B - I am willing to bet that most of you that read this far down before thinking to hell with it will be thinking what the hell does C(50,3) = 50!/3!47! mean. C(50,3) is a short way of writing the possible combinations of 3 cards drawn from a deck of 50. And 50!/3!47! is basically the calculation that is used written in a shorter way. The '!' means Factorial, a factorial of any whole number is found by multiplying all the positive number equal to or below that number, e.g. the factorial of 3 is 1 x 2 x 3 = 6, while the factorial of 5 is 1 x 2 x 3 x 4 x 5 = 120. The sign is used to save time and in a lot case paper, because in this calculation we need the factorial of 50, which would take up a hell of lot of paper seeing as the answer to it is 65 numbers long.
So in order to calculate the possible number of flops we are choosing 3 cards from 50, when the numbers are put into the calculation we 50!/3!47! = 19,600. All the calculations for set of probablilites are done in the same way obviously as the pocket pair gets the lower the possible flops with overcards gets higher.

[Terrier01]

I was never very good with maths, but undertand your post.

I tend to overlook the lower pairs due to being a believer in the Sklansky hand rankings.

I play many SnGs and use them to great effect, it is worth looking on the internet for the breakdown of hands, some of which are contentious, but most are in agreement that the top 10 hands will normally offer good winning potential.

Betting the hands correctly and having the ability to fold when overcards, flushes or straights are evident on the flop.

It all comes down to the simple facts, does he/she have what I think is a better or worse hand?

Watch how the betting developes, don't get distracted and fold a lot.

[madjek]

Though I'd look at old threads, since statistics interests me, this one jumped out. A simpler way to calculate these odds are to just look at odd of not hitting over card on each street then multiplying odds together for streets that you're looking at. To use his example, QQ, to not hit over card on 1st street is 42/50 2nd 41/49 3rd 40/48. So chances of not hitting over card on flop would be (42*41*40)/(50*49*48) = .586. So chances of an over card on flop would be 1 - .586 = .414 or 41.4%, same results without complicated math. Problem with these odds is they are only accurate if opponents don't have an over card. For example, if you have QQ and opponent has A K, his odds of hitting over card on flop are not 41.4%. Then the odds to not hit would be 42/48 * 41/47 * 40/46 = 66-67% so odds of A or K hitting flop would be 33-34%