Hold'em or Fold'em?: Simulating Poker Deals

Introduction

If you are like me, you know enough about playing Hold'em to have a good time with friends, but not enough to win consistently or to bet competently.  Looking at strategies and tips for betting left me a little confused, so I turned to what I know to help me: simulation.

Through the use of simulated poker deals, I figured I could derive some principles for how to understand the strength of my hand in any given deal.  And whether or not I should hold'em or fold'em.

Obviously, the following has nothing to say about bluffing, or when or what to bet.  I merely offer some observations about which hands are likely to win based on the probabilities of hand distributions.  You can use these observations to develop your own strategies for how to win.

Hand Distributions

The following is a table of the distribution of hands over 100,000 simulated deals for a single player.  That is, it contains the likelihood that you will get a particular kind of hand on any given deal with your two dealt cards in combination with the five community cards.

Player1	Count	Percentage
9. One Pair	44239	44.24
8. Two Pair	23279	23.28
10. High Card	17791	17.79
7. Three of a Kind	4918	4.92
6. Straight	3968	3.97
5. Flush	2821	2.82
4. Full House	2642	2.64
3. Four of a Kind	174	0.17
2. Straight Flush	145	0.15
1.Royal Flush	23	0.02

I ran this simulation several times and the results were similar each time.  What I found most interesting is that it is less likely to only have a high card hand than it is to have a one pair or a two pair hand.  This is surprising because a high card is ranked lower than a one pair or a two pair hand.  One would expect that the ranking of hands would follow their likelihood of occurring.  This is the case for all hands, except for the single high card hand.

What does this mean practically? Well, one has 68% chance of getting a one pair or two pair.  Each individually is more likely than a single high card hand.  Overall, one has an 82% chance of not having a single high card hand.  So it is unlikely that one will only have a high card hand after all of the cards are dealt.

Chances Against One Other Player

Below is a table of how often Player1 won, tied, or lost whenever Player1 had the specified hand dealt.  This is over 100,000 simulations.  Note that I do not use tie-breaking rules to determine a final winner.  Instead, a tie designates that Player1 and Player2 had the same kind of hand (although one or the other may actually have won given the tie breaking rules).  I do this to simplify this initial round of analysis.  Also note that when I use the word "win", I am excluding ties in the calculation (see below), as I am concerned with wins versus losses.

Player1	Lose	Tie	Win	Win/(Win+Loss)	Tie/All
1.Royal Flush	0	0	23	1.00	0.00
2. Straight Flush	0	5	140	1.00	0.03
3. Four of a Kind	0	24	150	1.00	0.14
4. Full House	40	409	2193	0.98	0.15
5. Flush	80	620	2121	0.96	0.22
6. Straight	150	768	3050	0.95	0.19
7. Three of a Kind	733	1001	3184	0.81	0.20
8. Two Pair	3274	8857	11148	0.77	0.38
9. One Pair	14830	20968	8441	0.36	0.47
10. High Card	11655	6136	0	0.00	0.34

Some observations:

• A Royal flush, straight flush, and four of a kind always won or tied.  There is practically 0% chance of losing.
• When calculating win/(win+loss), a full house, flush, and straight won 98%, 96%, and 95% of the time, respectively.  In other words, there is a very small chance of losing.  One is more likely to tie than to lose (15%, 22%, and 19% chance of tying, respectively).
• When calculating win/(win+loss), three of a kind and two pair won 81% and 77% of the time, respectively.  Chances of tying are 20% and 38%, respectively.  One has a small chance of losing.  One is more likely to win or tie.
• When calculating win/(win+loss), one pair won only 36% of the time.  One pair tied 47% of the time.
• A high card always loses or ties.  Chances are 34% for a tie.

Given the foregoing, it is reasonable to suggest that if one has a two pair or more, one should stay in play.  Why?  Because chances are that one has the best hand.  Of course, you probably want to have a high two pair (AA or KK) and don't want that high pair in the community cards.  But overall, if you have a two pair or better, you should stay in. 

On the other hand, if you have a one pair or high card only, you should fold (or be really good at bluffing).  You are more likely to lose (or tie) than win.

Chances Against Two Other Players

Things change a little with two other players.  Here is the table of how often Player1 won, tied, or lost when two other players were in the game:

Player1	Lose	Tie	Win	Win/(Win+Loss)	Tie/All
1.Royal Flush	0	0	33	1.00	0.00
2. Straight Flush	0	5	150	1.00	0.03
3. Four of a Kind	0	25	156	1.00	0.14
4. Full House	31	765	1684	0.98	0.31
5. Flush	88	647	2220	0.96	0.22
6. Straight	170	777	3123	0.95	0.19
7. Three of a Kind	1538	107	3238	0.68	0.02
8. Two Pair	3338	16649	3439	0.51	0.71
9. One Pair	24953	19101	0	0.00	0.43
10. High Card	17763	0	0	0.00	0.00

Some observations:

• A Royal flush, straight flush, and four of a kind always won or tied.  There is practically 0% chance of losing.
• A full house, flush, and straight won 98%, 96%, and 95% of the time, respectively.  So very little has changed in that regard.  However, one is much more likely to tie now in a full house (31%).
• Three of a kind and two pair won only 68% and 51% of the time, respectively.  Chances of tying go down to 2% and up to 71%, respectively.  These hands are much less likely to win now.
• One pair won 0% of the time and tied 43% of the time.
• A high card always loses and never ties.

With the addition of one more player, the higher ranking hands (1-6) change very little.  However, three of a kind and two pair are no longer largely dependable for a win.  And a one pair or high card are guaranteed losses.  Given this, you need a straight or better for a relatively sure thing.  Be prepared for bluffing and a uneasy fight if you only have two pair or a three of a kind.

Chances Against Five Other Players

Finally, consider a game starting with 6 players total.  What does Player1 need now to have the best hand among the group?  Here is the table of how often Player1 won, tied, or lost when five other players were in the game:

Player1	Lose	Tie	Win	Win/(Win+Loss)	Tie/All
1.Royal Flush	0	1	21	1.00	0.05
2. Straight Flush	0	6	153	1.00	0.04
3. Four of a Kind	0	24	157	1.00	0.13
4. Full House	43	848	1723	0.98	0.32
5. Flush	105	643	2088	0.95	0.23
6. Straight	166	771	3040	0.95	0.19
7. Three of a Kind	1676	99	3054	0.65	0.02
8. Two Pair	3289	16993	3442	0.51	0.72
9. One Pair	25147	19029	0	0.00	0.43
10. High Card	17482	0	0	0.00	0.00

Some observations:

• A Royal flush, straight flush, and four of a kind always won or tied.  There is practically 0% chance of losing.  This is largely the same.
• A full house, flush, and straight won 98%, 95%, and 95% of the time, respectively.  So this is practically the same. One's chances of tying with a full house are about the same too (32%).
• Three of a kind and two pair remain practically the same, 65% and 51%, respectively.  Chances of tying are also the same.
• One pair stays the same.
• A high card stays the same.

This is a very interesting result.  I would have thought that as the number of players increases, one's chances of losing increase, and hence, the need for an even better hand to win.  But the probabilities are practically the same against two players as against five players.  The advice stays the same.

 

Conclusion

In summary, here are some general takeaways:

• Expect at least a pair to have been dealt to you after all community cards have been dealt.
• Against one other player, (ignoring ties) two pair is generally good enough to win (79%).  Anything above and including a straight is practically guaranteed to win.
• Against two to five other players, (ignoring ties) expect to win half the time with two pair, two-thirds of the time with three of a kind, and almost all the time with a straight and above.

Good luck!