Poker: Probabilities of the Various Hands and TexasHold’em

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Some General Ideas About Counting

Last time we computed the probability of getting a royal and straightflush by writing out all possibilities. This was fine for these hand butisn’t a good method in general.

In order to compute the probability of other hands, one approach is todecide what things you need to choose in order to write down a hand,and then determine how many ways each of the choices can occur.

For example, to count royal flushes, you only have to choose a suitbecause we must then take 10, J,Q,K ,A of that suit.

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Counting Independent Events

If one event does not affect the outcome of another, they are calledindependent. To count the number of ways a pair of independentevents can occur, multiply the number of ways each way can occur.

This is basically the same mathematical idea as multiplyingprobabilities in a probability tree we used last week.

Rolling two dice is an example of two independent events: what youget on one die does not affect what can happen on the other. Sincethere are 6 outcomes for rolling one die, there are 6 · 6 = 36 outcomesfor rolling two dice.

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Probability of a Straight Flush, Revisited

How can we use the idea above to count the number of straightflushes?

To choose a given straight flush, you must choose a suit, and astarting (or ending) value for the 5 in a row. There are 4 choices forthe suit.

Q How many ways are there to choose the starting value of a straightflush (which is not a royal flush)? Enter the number with your clicker.

A There are 9 values. It would appear that there are 10 possible startingvalues (A through 10). However, if we want a straight flush which isnot a royal flush, we cannot start at 10, so there are 9 choices.

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Choosing the suit and the starting value are independent events. So,to count the number of straight flushes, we need to multiply thenumber of choices of suit and the number of choices of starting value.

Therefore, there are 4 · 9 = 36 total straight flushes.

The probability of a straight flush (which is not a royal flush) is then36/2,598,960, which is about 1 out of 72,000.

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4 of a Kind

What is the probability of a 4 of a kind? An example is

8♠, 8♥, 8♦, 8♣, J♥

To get a 4 of a kind, you must choose the value of the 4 of a kind,and choose the remaining card.

There are 13 choices for the value of the 4 of a kind.

Q How many choices are there for the remaining card?

A 48. The 5th card can be any of the remaining 48 cards.

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Because we can view writing down a 4 of a kind as choosing the valueof the 4 of a kind and picking the last card, which are independentevents, we need to multiply the number of choices together.

The number of four of a kinds is then 13 · 48 = 624. The probabilityof a 4 of a kind is then 624/2598960, or about 1 out of 4200.

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Full House

A full house consists of a 3 of a kind and a 2 of a kind. What is theprobability of getting a full house? An example is

4♥, 4♦, 4♣, J♠, J♥

To have a full house you must choose the value of a 3 of a kind andthe value of a 2 of a kind. You must also choose which 3 cards makeup the 3 of a kind and which 2 make up the 2 of a kind. This is oneof the more complicated counts.

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Clicker Questions

Q How many ways are there to choose the value of the 3 of a kind?

A There are 13C1 = 13 ways to choose the value of the 3 of a kind.

Q How many ways area there to choose the value of the 2 of a kind?

A There are 12C1 = 12 ways to choose the value of the pair. You canchoose any value other than the value of the 3 of a kind.

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More Clicker Questions

Q For sake of argument, say we choose Jacks for the 3 of a kind and 7for the 2 of a kind. How many ways are there to choose the 3 jacks?

A There are 4C3 = 4 ways to choose the 3 Jacks. They are

J♠J♥J♦ J♠J♥J♣ J♥J♦J♣ J♠J♦J♣

Q How many ways are there to choose the two 7s?

A There are 4C2 = 6 ways to choose the two 7s. They are

7♠7♥ 7♠7♦ 7♠7♣ 7♥7♦ 7♥7♣ 7♦7♣

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To find the number of full houses, we then have to multiply thenumbers of our various choices (value of the 3 of a kind, value of thepair, the three cards for the 3 of a kind, the two cards for the pair).

So, the number of full houses is

13 · 12 · 4 · 6 = 3744

The probability of a full house is then 3744/2598960, which is about1 out of 4000 hands.

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Probability of a Flush

In order to have a flush, which is 5 cards of the same suit, we need tochoose the suit, and then pick the five cards. There are 4 choices forthe suit.

Q Suppose we decide to get a flush with hearts. How many ways arethere to pick 5 hearts?

A We need to pick 5 of the 13 hearts. The number of ways is 13C5.This is equal to 1287.

The number of flushes is then 4 · 1287 = 5148. However, this includesroyal and straight flushes. Subtracting the 40 of them gives 5108flushes. This means the probability of being dealt a flush is5108/2598960, or about 1 out of every 500 hands.

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Probabilities of the Various Hands

We could continue with the remaining types of hands. But, insteadwe summarize the results:

Type of Hand Number of that Type PercentageRoyal Flush 4 0.00015%

Straight Flush 36 0.001%

4 of a Kind 624 0.02%

Full House 3,744 0.15%

Flush 5,108 0.2%

Straight 10,200 0.4%

3 of a Kind 54,912 2.1%

2 Pair 123,552 4.8%

1 Pair 1,098,240 42.3%

Nothing 1,302,540 50.1%

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Texas Hold’em

Let’s start with a YouTube claymation video of Homer and Bartplaying Texas Hold’em.

Simpson’s Video

Here is a video of Lisa playing Hold’em.

www.pokertube.com/videos/the-simpsons-online-poker-episode

Finally, here is the rest of the video we started to watch last time:

How to play: Texas Holdem Poker

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How is Texas Hold’em Played

• Two cards are dealt to each player. A round of betting ensues.

• Three cards are then dealt face up (the flop). These are communitycards; anybody can use them. Another round of betting is done.

• Another card is dealt face up (the turn). A round of betting ensues.

• A final card is dealt face up (the river). The final round of bettingoccurs.

• Of the players that have not folded, the player with the best 5-cardhand, made from any combination of his two cards and the cards onthe board, wins.

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Number of 2 Card Hands

You start Hold’em by being dealt a 2 card hand. How many different2 card hands are there?

Since you get 2 cards from the 52 card deck, then number is 52C2,which is equal to 1,326.

We’ll use this to compute the probability of being dealt a pair withyour two cards and, if time permits, the probability of getting 2 cardsof the same suit.

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Number of Ways to Deal the 5 Cards Face Up

Q How many ways are there to deal the 5 cards face up? It involves abinomial coefficient nCr . The value of r is 5. What is the value of n?

A Since you are seeing 2 of the cards, there is only 50 cards to choosefrom, so the number of ways is 50C5, which is equal to 2,118,760.

If we are only interested in how many ways we can flop 3 cards, thenumber is 50C3 = 19,600.

We’ll use this to calculate the probability of ending up with a givenhand in Hold’em.

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Probability of Being Dealt a Pocket Pair

What is the probability you are dealt a pair? This is called a pocketpair.

In order to write down all possible pairs, we need to choose the valueof the pair, then choose the actual two cards.

There are 13 choices for the value of the pair.

Q Once we pick the value, how many choices are there for the two cardsin the pair?

A There are 4C2 = 6 choices for the two cards in the pair.

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The total number of ways to be dealt a pair is then 13 · 6 = 78.

Recall that the total number of 2-card hands you can be dealt is

52C2 = 1326.

Then the probability of being dealt a pocket pair is 78/1326, or 1 outof 17 hands.

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Being Dealt Suited Cards

If you are interested in flushes, you’d like to know how likely it is thatyou are dealt two cards of the same suit to begin the game.

To be dealt 2 of the same suit, you much choose the suit and thenchoose the cards of that suit.

There are 4 choices for the suit.

Q How many choices are there to pick the 2 cards of the suit? It is nC2

for some n. What is n?

A n = 13 because we are choosing from the 13 cards of the suit. Thereare 13C2 = 78 ways to pick 2 cards of the suit.

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We’ve seen that there are 4 choices for the suit and 78 choices forpicking the 2 cards of the suit.

We’ve also seen that there are 52C2 = 1, 326 ways to pick 2 cardsfrom the deck.

The total number of ways of getting 2 of the same suit is then4 · 78 = 312. Then the probability of getting 2 of the same suit is312/1326, or a little worse than 1 out of 4 hands.

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Hitting Three of a Kind on the Flop

Suppose you have a pocket pair. What is the probability that you’llget 3 of a kind on the flop?

Let’s say we have 2 Aces. In order to get 3 of a kind on the flop (thefirst three cards dealt up), we need to have one of the three cards bean Ace and the other two anything else. We need to select an Aceand select two other cards.

There are 2C1 = 2 ways to select one of the remaining Aces.

There are 48C2 ways to select two non-Aces. This number is equal to(48 · 47)/2 = 1128.

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There are then 2 · 1128 = 2256 ways for this to happen. There are

50C3 = 19, 600 total ways to select 3 cards.

The probability of hitting 3 of a kind on the flop is then 2256/19600,which is about 1 time out of 9, or a little more than 10% of the time.

This does include getting 4 of a kind on the flop, but that happensrarely, so the probability of getting exactly 3 of a kind is almost whatwe just calculated.

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Hitting 3 of a Kind with a Pocket Pair

If you have a pocket pair, what is the probability of hitting 3 of a kindby the end of the hand?

We’ve seen that the number of ways to deal the five face-up cards is

50C5 = 2,118,760.

If we have, say, a pair of Aces, to end up with 3 of a kind, of the fivedealt cards one must be an Ace and the other 4 anything else.

To get this we must choose an Ace and choose 4 other cards. Thereare 2C1 = 2 ways to pick the Ace.

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Q How many ways are there to pick the 4 other cards? It is nC4 forsome value of n. What is the value of n?

A n = 48. There are 48C4 = 194,580 ways to pick the other cards.

Then 2 · 194,580 = 389,160 total ways for this to happen.

The probability of hitting 3 of a kind is then 389, 160/2,118,760,which is about 1 in 5.

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Hitting a Flush

Suppose you are dealt two spades. What is the probability you’ll endup with a flush?

In order to hit a flush, we must have 3 spades dealt along with 2other cards.

There are 11C3 = 165 ways to pick 3 of the 11 spades remaining (2are in our hand).

There are 47C2 = 1081 ways to pick 2 more cards. We can pick morespades and still end up with a flush.

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The total number of ways the 5 cards can be dealt and we end upwith a flush is then 165 · 1081 = 178,365.

The total number of ways the 5 cards can be dealt is

50C5 = 2,118,760.

The probability of hitting a flush is then 178,365/2,118,760, or about1 in every 12 hands that we are dealt two spades.

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Hitting a Runner Runner Flush

Suppose after the flop, you have 3 spades. What is the probabilityyou’ll get a flush?

You need to get at two spade on the last two cards. At this point yousee 5 cards, the 2 in your hand and the three on the board. So, thenumber of ways of dealing the final two cards is 47C2 = 1081.

In order to get two spades on the last two cards, you need to pick twoof the 10 spades. Why 10? You are seeing 3, so there are 10remaining to select. Then the number of ways of picking 2 of them is

10C2 = 45.

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The probability of hitting a flush when you have 3 spades at the flopis then 45/1081, which is about 1 in 24 hands, or about 4%. Not toooften!

Trying to get a flush is a desirable thing, but this doesn’t happenoften, so you probably don’t want to chase flushes in this way.

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Hitting a Gut Shot

Suppose after the flop you have 4 of 5 cards needed to make astraight, but you are missing a card in the middle. For example, youmight have 4♠, 6♥, 7♠, 8♦.

What is the probability of getting the straight? This is also called aninside straight.

There are 47C2 = 1081 ways to deal the last two cards. In order to hityour straight you must get a 5 and another card. There are 4 choicesfor the 5, and 46 choices for the remaining card.

There are 4 · 46 = 182 total ways to end up with your straight, so theprobability is 182/1081, or about 1 out of 6.

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Hitting an Outside Straight

Suppose after the flop you have 4 of 5 cards needed to make astraight, and they are consecutive. For example, suppose you have8♣, 9♥, 10♣, J♠. What is the probability of hitting the straight?

Q Of the remaining cards in the deck, how many will give you astraight?

A 8. You must hit either a 7 or a Queen, and there are 4 of each.

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There are 8 choices for the 7 or Queen, and 46 choices for theremaining card.

We have 46 because we see two in our hand and 4 on the board.

There are 8 · 46 = 364 ways to hit your straight. Since there are 1081total ways to deal the last two cards, so your probability is 364/1081,or 1 in 3. This is twice as likely as hitting an inside straight.

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Next Week

We will finish up our discussion about Hold’em. We’ll also see aformula for expected value. Then we will discuss the odds of winningvarious New Mexico Lottery games. We’ll use the notion of expectedvalue quite a bit to understand how much money the state can expectto receive from people playing the lottery. We’ll also see otherapplications of expected value.

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