Quote:
Originally Posted by _GUN_  I wrote down my numbers from a tournament today.
58 hands, saw 7% of the flops, won 0 pots, saw 1 turn, 0 pre-flop raises
in 58 hands, I was dealt 0 pairs and one ace (A8o). A8o came on hand four and the pot had already been opened, so I dumped it.
what are the odds on zero pairs and one ace in 58 hands? |
There are 1326 possible starting hands. (52 * 51) / 2 ; we divide by 2 because (for example) AhKd is the same hand as KdAh
Of those 1326 starting hands, 78 of them are pocket pairs. There are 6 ways to make any given pocket pair (e.g., AsAh AsAd AsAc AhAd AhAc AdAc) and there are 13 types of pairs. 6 * 13 = 78.
Besides the 6 pocket aces, there are 4 * 48 = 192 other hands that include an Ace.
Given all that, we can compute what you ask, except first we need to figure out exactly what you are asking.
To make it simple, let's just ask the question: what is the probability of going N hands without seeing either a pocket pair or any type of Ace?
The odds of getting a not-pair-not-Ace (I'll call this NPNA) hand is simply the number of such hands divided by 1326 (total number of possible hands).
There are 1056 NPNA hands ( = 1326 - (78 + 192) )
Probability of 1 NPNA hand = 1056/1326 or about 79.6%
The probability of getting N NPNA hands is simply (1056/1326) ** N
So if you want to know (for rough estimation purposes) what the odds of getting 50 NPNA hands in a row are, they are ( (1056/1326) ** 50 ) which works out to about 0.00001138 or about 88,000:1 against.
Figuring out the odds of getting exactly 1 not-NPNA hand and 58 NPNA hands, with the one "good" hand coming anywhere (not necessarily 4th hand) is left as an exercise for the reader. Suffice to say, it's not likely, as the already-low "get 50 NPNA hands in a row" probability suggests.
That said, I think it's important to point out that highly unlikely things happen every single time you play poker. Take a deck of cards, shuffle them, and then look at them. The odds that those cards will be in THAT order (whatever order they happen to be in) are 1 in 52 factorial, which is astronomically unlikely.
And yet, THERE THEY ARE ... in spite of how unlikely it is that they will be in that particular order.
In other words, once it happened to you, it's no longer unlikely. And given how many poker sessions are run every day in the world, it's not surprising that someone runs into that "one in a few hundred thousand" rarity.
Well, that's enough math for one night, yes?
