I was never been good at those type of endeavors. In all honesty, nearly ten years of post secondary education and I have no idea of how to go about changing the oil in mah karr. Nonetheless, I have developed a series of humble abilities, which enable me to answer a few of the questions that have bugged me since my ex gf introduced me to poker.
The other day, after reading a bit about the psychology involved in decision making, I was relating to a specific idea in the book that stated that people tend to overestimate small probabilities (see Prospect Theory). For instance, the other day I was in a casino in Sweden and I was freaking out at other people having better hole cards than I, and this was happening all.the. fucking time!
So I decided to write a small MACRO to estimate via monte carlo simulations the probability of one to more people having cards that add up to 21 (i.e. a ten with some other face card and an ace) or a pair. Did this really happen as frequently as I was assuming in the cash game back in Goeteborg, or was I suffering from this documented, overestimation of small-probability events?
I went about having the computer be mah beetch and figure it out by simulating around 1,000,000 games of poker, preflop, and to evaluate the type of cards that are handed to each player. The results are summarized below.
The first table shows the approximate probability that the cards handed in will either be a pair or two hole cards that add up to 21 or more, each row is for a certain number of players: the first row for a game of 2 players, the second row for 3, etc. Column-wise, the table shows by number of players having such cards: first column is just one or more players from the total number of people in the game, the second column two or more players from the total number of people in the game, etc...
The second table does the same thing with the sole difference that the probabilities are just for PAIRS, hence the smaller values.
| Pair of Cards adding 21 | ||||
| 1 or more | 2 or more | 3 or more | 4 or more | |
| 2 | 0.5382 | 0.0924 | 0 | 0 |
| 3 | 0.6815 | 0.2352 | 0.0281 | 0 |
| 4 | 0.7874 | 0.374 | 0.0868 | 0.0072 |
| 5 | 0.8605 | 0.5071 | 0.168 | 0.027 |
| 6 | 0.9084 | 0.6253 | 0.2734 | 0.0728 |
| 7 | 0.9355 | 0.7089 | 0.3759 | 0.1276 |
| 8 | 0.9632 | 0.7958 | 0.4899 | 0.2034 |
| 9 | 0.9758 | 0.8523 | 0.5916 | 0.2973 |
| 10 | 0.985 | 0.8949 | 0.6743 | 0.3839 |
| Pairs Only | ||||
| 1 or more | 2 or more | 3 or more | 4 or more | |
| 2 | 0.1035 | 0.0029 | 0 | 0 |
| 3 | 0.1569 | 0.008 | 0.0001 | 0 |
| 4 | 0.206 | 0.0174 | 0.0006 | 0 |
| 5 | 0.2395 | 0.0256 | 0.0015 | 0 |
| 6 | 0.2914 | 0.0369 | 0.0025 | 0.0002 |
| 7 | 0.3196 | 0.0506 | 0.0039 | 0.0005 |
| 8 | 0.3565 | 0.0686 | 0.0083 | 0.0006 |
| 9 | 0.3972 | 0.0811 | 0.0123 | 0.0009 |
| 10 | 0.4329 | 0.1001 | 0.0153 | 0.0019 |
From these values, which are the important ones? If you play cash games, I venture to say that rarely will there be more than 7 people in the table, and any other number below 5% should be considered as highly unlikely; nonetheless, they do tend to happen, so always watch out for position, bet size vs. pot, and all that other shit that one must pay attention to.
So, what is the proabability that 3 or more players will have good cards as defined before? In a cash game with only 7 players, this comes out to 37.59%, if you only focus on pairs, the value decreases to 00.39%! The first number is surprisingly bigger than what I had anticipated, whereas the second one is way smaller than what I would've assumed. So I did fall into that psychological trap-dammit!
I will continue to answer questions like this throughout the summer, as to not die from utter boredom. Namaste.
