Poker: Probabilities of the Various Hands and Texas Hold em. Poker 2 1/33

Transcription

1 Poker: Probabilities of the Various Hands and Texas Hold em Poker 2 1/33

2 Some General Ideas About Counting Last time we computed the probability of getting a royal and straight flush by writing out all possibilities. This was fine for these hand but isn t a good method in general. In order to compute the probability of other hands, one approach is to decide 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 suit because we must then take 10, J, Q, K, A of that suit. Poker 2 2/33

3 Counting Independent Events If one event does not affect the outcome of another, they are called independent. To count the number of ways a pair of independent events can occur, multiply the number of ways each way can occur. This is basically the same mathematical idea as multiplying probabilities in a probability tree we used last week. Rolling two dice is an example of two independent events: what you get on one die does not affect what can happen on the other. Since there are 6 outcomes for rolling one die, there are 6 6 = 36 outcomes for rolling two dice. Poker 2 3/33

4 Probability of a Straight Flush, Revisited How can we use the idea above to count the number of straight flushes? To choose a given straight flush, you must choose a suit, and a starting (or ending) value for the 5 in a row. There are 4 choices for the suit. Q How many ways are there to choose the starting value of a straight flush (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 starting values (A through 10). However, if we want a straight flush which is not a royal flush, we cannot start at 10, so there are 9 choices. Poker 2 4/33

5 Choosing the suit and the starting value are independent events. So, to count the number of straight flushes, we need to multiply the number 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 then 36/2,598,960, which is about 1 out of 72,000. Poker 2 5/33

6 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. Poker 2 6/33

7 Because we can view writing down a 4 of a kind as choosing the value of the 4 of a kind and picking the last card, which are independent events, we need to multiply the number of choices together. The number of four of a kinds is then = 624. The probability of a 4 of a kind is then 624/ , or about 1 out of Poker 2 7/33

8 Full House A full house consists of a 3 of a kind and a 2 of a kind. What is the probability 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 and the value of a 2 of a kind. You must also choose which 3 cards make up the 3 of a kind and which 2 make up the 2 of a kind. This is one of the more complicated counts. Poker 2 8/33

9 Clicker Questions Q How many ways are there to choose the value of the 3 of a kind? A There are 13 C 1 = 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 12 C 1 = 12 ways to choose the value of the pair. You can choose any value other than the value of the 3 of a kind. Poker 2 9/33

10 More Clicker Questions Q For sake of argument, say we choose Jacks for the 3 of a kind and 7 for the 2 of a kind. How many ways are there to choose the 3 jacks? A There are 4 C 3 = 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 4 C 2 = 6 ways to choose the two 7s. They are Poker 2 10/33

11 To find the number of full houses, we then have to multiply the numbers of our various choices (value of the 3 of a kind, value of the pair, the three cards for the 3 of a kind, the two cards for the pair). So, the number of full houses is = 3744 The probability of a full house is then 3744/ , which is about 1 out of 4000 hands. Poker 2 11/33

12 Probability of a Flush In order to have a flush, which is 5 cards of the same suit, we need to choose the suit, and then pick the five cards. There are 4 choices for the suit. Q Suppose we decide to get a flush with hearts. How many ways are there to pick 5 hearts? A We need to pick 5 of the 13 hearts. The number of ways is 13 C 5. This is equal to The number of flushes is then = However, this includes royal and straight flushes. Subtracting the 40 of them gives 5108 flushes. This means the probability of being dealt a flush is 5108/ , or about 1 out of every 500 hands. Poker 2 12/33

13 Probabilities of the Various Hands We could continue with the remaining types of hands. But, instead we summarize the results: Type of Hand Number of that Type Percentage Royal Flush % Straight Flush % 4 of a Kind % Full House 3, % Flush 5, % Straight 10, % 3 of a Kind 54, % 2 Pair 123, % 1 Pair 1,098, % Nothing 1,302, % Poker 2 13/33

14 Texas Hold em Let s start with a YouTube claymation video of Homer and Bart playing Texas Hold em. Simpson s Video Here is a video of Lisa playing Hold em. Finally, here is the rest of the video we started to watch last time: How to play: Texas Holdem Poker Poker 2 14/33

15 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 community cards; 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 betting occurs. Of the players that have not folded, the player with the best 5-card hand, made from any combination of his two cards and the cards on the board, wins. Poker 2 15/33

16 Number of 2 Card Hands You start Hold em by being dealt a 2 card hand. How many different 2 card hands are there? Since you get 2 cards from the 52 card deck, then number is 52 C 2, which is equal to 1,326. We ll use this to compute the probability of being dealt a pair with your two cards and, if time permits, the probability of getting 2 cards of the same suit. Poker 2 16/33

17 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 a binomial coefficient n C r. 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 choose from, so the number of ways is 50 C 5, which is equal to 2,118,760. If we are only interested in how many ways we can flop 3 cards, the number is 50 C 3 = 19,600. We ll use this to calculate the probability of ending up with a given hand in Hold em. Poker 2 17/33

18 Probability of Being Dealt a Pocket Pair What is the probability you are dealt a pair? This is called a pocket pair. In order to write down all possible pairs, we need to choose the value of 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 cards in the pair? A There are 4 C 2 = 6 choices for the two cards in the pair. Poker 2 18/33

19 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 52C 2 = Then the probability of being dealt a pocket pair is 78/1326, or 1 out of 17 hands. Poker 2 19/33

20 Being Dealt Suited Cards If you are interested in flushes, you d like to know how likely it is that you 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 then choose 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 n C 2 for some n. What is n? A n = 13 because we are choosing from the 13 cards of the suit. There are 13 C 2 = 78 ways to pick 2 cards of the suit. Poker 2 20/33

21 We ve seen that there are 4 choices for the suit and 78 choices for picking the 2 cards of the suit. We ve also seen that there are 52 C 2 = 1, 326 ways to pick 2 cards from the deck. The total number of ways of getting 2 of the same suit is then 4 78 = 312. Then the probability of getting 2 of the same suit is 312/1326, or a little worse than 1 out of 4 hands. Poker 2 21/33

22 Hitting Three of a Kind on the Flop Suppose you have a pocket pair. What is the probability that you ll get 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 (the first three cards dealt up), we need to have one of the three cards be an Ace and the other two anything else. We need to select an Ace and select two other cards. There are 2 C 1 = 2 ways to select one of the remaining Aces. There are 48 C 2 ways to select two non-aces. This number is equal to (48 47)/2 = Poker 2 22/33

23 There are then = 2256 ways for this to happen. There are 50C 3 = 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 happens rarely, so the probability of getting exactly 3 of a kind is almost what we just calculated. Poker 2 23/33

24 Hitting 3 of a Kind with a Pocket Pair If you have a pocket pair, what is the probability of hitting 3 of a kind by the end of the hand? We ve seen that the number of ways to deal the five face-up cards is 50C 5 = 2,118,760. If we have, say, a pair of Aces, to end up with 3 of a kind, of the five dealt 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. There are 2 C 1 = 2 ways to pick the Ace. Poker 2 24/33

25 Q How many ways are there to pick the 4 other cards? It is n C 4 for some value of n. What is the value of n? A n = 48. There are 48 C 4 = 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. Poker 2 25/33

26 Hitting a Flush Suppose you are dealt two spades. What is the probability you ll end up with a flush? In order to hit a flush, we must have 3 spades dealt along with 2 other cards. There are 11 C 3 = 165 ways to pick 3 of the 11 spades remaining (2 are in our hand). There are 47C 2 = 1081 ways to pick 2 more cards. We can pick more spades and still end up with a flush. Poker 2 26/33

27 The total number of ways the 5 cards can be dealt and we end up with a flush is then = 178,365. The total number of ways the 5 cards can be dealt is 50C 5 = 2,118,760. The probability of hitting a flush is then 178,365/2,118,760, or about 1 in every 12 hands that we are dealt two spades. Poker 2 27/33

28 Hitting a Runner Runner Flush Suppose after the flop, you have 3 spades. What is the probability you ll get a flush? You need to get at two spade on the last two cards. At this point you see 5 cards, the 2 in your hand and the three on the board. So, the number of ways of dealing the final two cards is 47 C 2 = In order to get two spades on the last two cards, you need to pick two of the 10 spades. Why 10? You are seeing 3, so there are 10 remaining to select. Then the number of ways of picking 2 of them is 10C 2 = 45. Poker 2 28/33

29 The probability of hitting a flush when you have 3 spades at the flop is then 45/1081, which is about 1 in 24 hands, or about 4%. Not too often! Trying to get a flush is a desirable thing, but this doesn t happen often, so you probably don t want to chase flushes in this way. Poker 2 29/33

30 Hitting a Gut Shot Suppose after the flop you have 4 of 5 cards needed to make a straight, but you are missing a card in the middle. For example, you might have 4, 6, 7, 8. What is the probability of getting the straight? This is also called an inside straight. There are 47 C 2 = 1081 ways to deal the last two cards. In order to hit your straight you must get a 5 and another card. There are 4 choices for the 5, and 46 choices for the remaining card. There are 4 46 = 182 total ways to end up with your straight, so the probability is 182/1081, or about 1 out of 6. Poker 2 30/33

31 Hitting an Outside Straight Suppose after the flop you have 4 of 5 cards needed to make a straight, and they are consecutive. For example, suppose you have 8, 9, 10, J. What is the probability of hitting the straight? Q Of the remaining cards in the deck, how many will give you a straight? A 8. You must hit either a 7 or a Queen, and there are 4 of each. Poker 2 31/33

32 There are 8 choices for the 7 or Queen, and 46 choices for the remaining 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 1081 total 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. Poker 2 32/33

33 Next Week We will finish up our discussion about Hold em. We ll also see a formula for expected value. Then we will discuss the odds of winning various New Mexico Lottery games. We ll use the notion of expected value quite a bit to understand how much money the state can expect to receive from people playing the lottery. We ll also see other applications of expected value. Poker 2 33/33