Banana Poker is a cute and whimsical poker video game where players can choose to play either high / low card or Texas Holdem Poker against 5 other computer players. While we do not promote gambling with real money, this game can be useful for helping to teaching mathematical concepts like odds and probability along with the importance of exercising patience.
Children and parents can play this cartoon card card game by clicking in the window below.
Alternatively kids and adults can play this virtual texas holdem cartoon card game for free as a web application here.
Bet on hands you think you will win while folding on hands you think you are likely to lose.
High low is a much easier and less complex game than poker, so it makes sense to explain it first.
The following table highlights how the competing players were programmed to play.
Breaking things down more specifically…
The following table highlights the random odds of any hand type in a 7-card poker game.
|Poker Hand||Frequency||Probability||Cumulative||Odds against|
|Royal flush||4,324||0.0032%||0.0032%||30,939 : 1|
|Straight flush (excluding royal flush)||37,260||0.0279%||0.0311%||3,589.6 : 1|
|Four of a kind||224,848||0.168%||0.199%||594 : 1|
|Full house||3,473,184||2.60%||2.80%||35.7 : 1|
|Flush (excluding royal flush and straight flush)||4,047,644||3.03%||5.82%||32.1 : 1|
|Straight (excluding royal flush and straight flush)||6,180,020||4.62%||10.4%||20.6 : 1|
|Three of a kind||6,461,620||4.83%||15.3%||19.7 : 1|
|Two pair||31,433,400||23.5%||38.8%||3.26 : 1|
|One pair||58,627,800||43.8%||82.6%||1.28 : 1|
|No pair / High card||23,294,460||17.4%||100%||4.74 : 1|
|Total||133,784,560||100%||—||0 : 1|
Since players share cards on in Texas Holdem the odds of certain hand combinations are higher. For example, if the table is showing 3 8s then there is a higher chance of a competitor having 4 of a kind or a full house than whatever the random odds would be since they too have that 3 of a kind in their hand. The odds of two players both having a straight or flush would be elevated as well if many of the key cards that make up the combination are on the table.
Hand rankings have a lot of value for limit play, however in no limit play their value deteriorates quickly as the downside of “all in” is an exit from the game. Here are some notable ranking systems.
David Sklansky and Mason Malmuth created a diagram suggesting when hands should be played, with stronger hands having a lower number.
Similarly, Phil Hellmuth’s 2003 book Play Poker Like the Pros grouped hands by tiers.
|1||AA, KK, AKs, QQ, AK||Top 12 Hands|
|2||JJ, TT, 99||Top 12 Hands|
|3||88, 77, AQs, AQ||Top 12 Hands|
|4||66, 55, 44, 33, 22, AJs, ATs, A9s, A8s||Majority Play Hands|
|5||A7s, A6s, A5s, A4s, A3s, A2s, KQs, KQ||Majority Play Hands|
|6||QJs, JTs, T9s, 98s, 87s, 76s, 65s||Suited Connectors|
Bill Chen developed what he called the “Chen Formula” which assigns points as follows
Multiply the points by 2. A pair of aces would be worth 20, a pair of kings would be worth 16, and the minimum value for a pair is 5.
If cards are of the same suit add 2 points as you are more likely to be able to get a flush.
This is given weight based on the probability you may be able to get a straight.
After the flop you then have to shift your approach based on the cards on the table.
The strength of hands is as follows:
The approximate odds of various hands was shown above in the 7-card poker section, though the odds of rarer hands could technically be experienced more commonly in texas hold’em (particularly limit play) as players are not committed to throw away cards in order to see additional cards. A player could make a straight or a flush on the final card when they may not have played those cards in a 7-card game where they were forced to throw away a pair in order to make a flush or straight. Here are the odds using a 7 out of 52 hand where you get to make your 5 cards from a group of 7 you hold.
|Hand||Example Cards||Possibile Combos||Probability||Odds|
If two players have the exact same hand then the pot is split. If both hands are quite similar then the card value of the most important piece determines the winner. For example, if 2 players have a full house then whoever has the higher 3 of a kind wins. And if the 3 of a kind is the same for both players then whoever has the highest pair wins. For flushes or straights whoever has the highest card as part of that sequence wins. For games where both players have the same 4 of a kind, 3 of a kind, pair, etc. … then the remaining high card the player hold determines who wins.
Many great hands are made on the flop between the cards in player’s hands and the cards on the flop. The flop by itself typically does not have much in the way of playable cards in absolute isolation. Typically the flop by itself (without the cards in player’s hands) is rather pedestrian. On the flop most the time cards are not connected, no pairs are shown, and there are rarely more than 2 cards in the same suit.
|3 suited cards||5.17%||18.34:1|
|2 suited cards||55%||0.82:1|
|3 connected straight cards||3.45%||27.99:1|
|2 connected straight cards||40%||1.5:1|
|No connected cards||55.6%||0.799:1|
The following table shows your probability of various hand improvements on the flop given a specific set of pocket cards.
|Pocket hand||Flop Improvement||Probability||Odds|
|Pair||Three-of-a-kind or better||12.7%||6.9:1|
|2 unpaired cards||Pair||32.4%||2.1:1|
|2 unpaired cards||Two pair||2%||48.5:1|
|Suited cards||Flush draw||10.9%||8.17:1|
|Suited cards||Backdoor flush draw||41.6%||1.4:1|
|Connectors 45o-JTo||open ended straight draw||9.6%||9.42:1|
|Connectors 45s-JTs||Straight draw / flush draw
After the flop comes the turn. The following table shows the odds of upgrading your hand on the turn.
|Current Hand||Turn Goal||Probability||Odds|
|Open ended straight draw||Straight||17%||4.9:1|
|Gutshot straight draw||Straight||8.5%||10.76:1|
|Two unpaired cards||Pair with a hole card||12.8%||6.8:1|
After the turn comes the river. The following table shows the odds of upgrading your hand on the river.
|Current Hand||River Goal||Probability||Odds|
|Open ended straight draw||Straight||17.4%||4.74:1|
|Gutshot straight draw||Straight||8.7%||10.5:1|
|Unpaired cards||Pair with hole card||13%||6.7:1|
Each Texas Holdem hand has an “ideal” or best outcome available to it based on how things look after the flop.
After you have seen the flop and the cards in your hand you may need 1 card to complete a flush, a straight, a full house, a 3 of a kind, etc.
What you do is figure out how many cards from the deck remain and how many would satisfy your best hand.
For example, if you have 2 clubs in your hand and 2 clubs are showing on the table that means 4 of the 13 clubs are already gone, so there are 9 remaining clubs.
Each deck has 52 cards. You know what the 3 cards on the flop were as well as the 2 in your hand, so if you subtract those 5 cards that means there are 47 left. On the turn you would have a 9 in 47 (or 19.1%) chance of getting another club. If that card was not a club you would have a 9 in 46 (or 19.6%) chance on the river.
|Outs||Odds After Flop||Odds After Turn|
Below are the odds of completing common card scenarios on the turn, river, or across both.
|Hand||Outs||Flop to Turn||Turn to River||Turn + River||Odds|
|Pocket pair to 3 of a kind||2||4.3%||4.3%||8.4%||10.9:1|
|Pair to 4 of a kind||1*||–||–||0.09%||1,100:1|
|Completing a pair of a high card||3||6.4%||6.5%||12.5%||7:1|
|Completing a pair of either high card||6||12.8%||13.0%||25%||3:1|
|Inside (gunshot) straight draw to straight||4||8.5%||8.7%||16.5%||4.88:1|
|Two pair to full house||4||8.5%||8.7%||16.5%||4.88:1|
|Two overcards to overpair||6||12.8%||13%||24.1%||3.15:1|
|3 of a kind to 4 of a kind||1||2.1%||2.2%||4.3%||22.26:1|
|3 of a kind to full house or 4 of a kind||7||14.9%||15.2%||27.8%||2.60:1|
|Open ended straight draw to straight||8||17%||17.4%||31.5%||2.13:1|
|Flush draw to flush||9||19.1%||19.6%||35%||1.86:1|
|Backdoor flush draw to flush||1*||–||–||4.2%||22.8:1|
|Inside straight and flush draw to straight or flush||12||25.5%||26.1%||45%||1.22:1|
|Open ended straight and flush draw to straight or flush||15||31.9%||32.6%||54.1%||0.85:1|
* Rather than there being multiple outs, you need 2 consecutive cards to go your way. Individual odds not shown for flop and turn since you need both to go your way.
When deciding if it makes sense to play you not only have to consider the odds of victory, but also the size of the pre-existing pot and the amount your competitor bet. If the raise was small relative to the pot size then you have to factor in the larger potential upside of winning the larger pot against the downside of the smaller incremental raise.
The above is purely mathematical in terms of considering probability on a particular bet. There are also human emotions, bluffing, re-raising a raise, and many other factors to consider which makes poker a complex game beyond the math aspect of pot odds.
Winning at poker is not only figuring your odds of hitting what you want, but also if an opponent have almost the same hand with a higher kicker or if the shared cards which enabled your great hand enabled some other better hand for them.
If you are holding 2 pocket spades and you get a flush on the turn or river and there are only a total of 3 spades showing on the table the odds of someone else having a spades flush are low.
If, however 4 spades are on the table and many players are still in the game it is likely one of the competing players also has a flush, so in that case it matters what spade you have in your hand. If you have a 2 or a 4 and they have a facecard you will lose.
Most players may bail on a potential flush if they only have 1 card in their hand, get 2 on the flop and need both the turn and river to be of that same suit. If, however, the flop itself has 3 of the same suit there are likely to be other players holding a spade who stayed in the game hoping on a fifth spade.
The above sort of thinking is also true with a straight. If you have an outside straight and your card is to the low side there is a good chance a competing player has an outside straight with a higher top card than you do.
The same sort of thinking is true if you flop a 3 of a kind where 2 of the pair are on the flop. Someone else playing may have the 4th of that card & they might also have a higher high card than you do or a full house with a higher second card. If you flop a 3 of a kind with all 3 being on the flop there is a good chance someone else will have a 4 of a kind or a full house with another pair.
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