What s Luck Got To Do With It?

Transcription

1 What s Luck Got To Do With It? Peter Brooksbank Department of Mathematics Bucknell University

2 The Gambler s Objective Always make bets at favourable odds: A casino offers to pay out $110 for any successful $100 bet on black or red at roulette. A bookmaker is offering long odds on a horse that you happen to know is flying in training.

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4 How many distinct poker hands are there? How many choices of 5 cards from a total of 52? First imagine choosing a sequence of 5 cards. There are 52 choices for the 1st card. There are 51 choices for the 2nd card, and so forth There are =311,875,200 such sequences. But it doesn t matter in what order the cards come! How many ways are there to reorder 5 cards? You can put the 1st card in any one of 5 positions. You can now put the 2nd card in any one of the remaining 4 positions, and so forth. There are =120 ways to reorder. We have therefore counted each poker hand 120 times. # poker hands =311,875, =2,598,960.

5 How many are Full Houses? First count the number of Full Houses of a certain type, say Queens over Fives. 4 ways to choose three Queens from four. 6 ways to choose two Fives from four ways to choose a <Q,5> Full House. Now how many types are there? There are 13 choices for the first part of the type and 12 choices for the second. Notice that order matters! Queens over Fives is a different Full House from Fives over Queens. There are 13 12=156 distinct types. # Full Houses = =3,744.

6 Poker Hand Frequencies Straight Flush 40 4-of-a-Kind 624 Full House 3,744 Flush 5,108 Straight 10,200 3-of-a-Kind 54,912 Two Pair 123,552 One Pair 1,098,240 Trash 1,302,540

7 Texas Hold Em Poker Each player is dealt 2 cards face down; these are called the hole cards. 5 community cards are dealt face up. Each player uses the two hole cards together with the 5 community cards to make the best 5 card poker hand. The first 3 community cards are turned over together: this is the flop. The 4th community card is called the turn. The last community card is called the river. There are four betting rounds: pre-flop, flop, turn and river. All bets go into the pot. If there is a showdown at the end, the player with the best 5 card hand wins the pot.

8 What is the correct play? You are in a hand with a single opponent. With one card to come the board is: The bet is $10. The pot is $70. You hold: What do you do? A. Fold B. Call C. Raise

9 Explanation There are 4 cards that will give you a winning straight: any one of the four Eights. You are unlikely to win if any other card hits. 46 unseen cards: 4 are good and 42 are bad. The odds are therefore 42-to-4 (10.5-to-1) against you making your straight. There is $70 in the pot and it costs you $10 to call and see the last card. Thus the pot odds are 7-to-1. This is not enough and you should fold.

10 What about this hand? You are in a hand with a single opponent. With one card to come the board is: The bet is $10. The pot is $70. You hold: What do you do? A. Fold B. Call C. Raise

11 Explanation The difference here is that there are 8 cards that will give you a winning straight: any one of four Sevens, or any one of four Queens. 46 unseen cards: 8 are good and 38 are bad. It is therefore 38-to-8 (4.75-to-1) against you making your straight. The pot odds are again 7-to-1. This is now good enough and you should call.

12 With Two Cards To Come You are in a hand with a single opponent. With two cards to come the board is: The bet is $5. The pot is $15. You hold: What do you do? A. Fold B. Call C. Raise

13 Explanation There are 9 cards that will give you the probable best hand, namely the nut flush. With two cards to come there is a 35% chance that one of these 9 cards will appear. Thus the odds are 65-to-35 (1.85-to-1) against you hitting your hand. The pot odds are 3-to-1. BUT it is likely that your opponent will bet again on the turn (this time for $10) if a heart does not come. So if you plan to see the hand through to the end, you are in fact betting $15 to win $25. Thus the effective pot odds are 25-to-15 (1.67-to-1) These odds are not good enough so you should fold.

14 With more players in the hand You are in a hand with three opponents. With two cards to come the board is: A bet of $5 has been made by one opponent and called by the other two. There is $35 in the pot. You hold: What do you do? A. Fold B. Call C. Raise

15 Explanation The pot odds are 7-to-1. The odds against you hitting a probable winner are much less. You cannot fold. There are 12 cards that give you the probable winner: 9 hearts and the 3 other jacks. With 2 cards to come you will hit this 45% of the time. You can interpret this as owning 45% of the eventual pot. In poker parlance this is your pot equity. With 4 active players, each bet you make represents only 25% of the total contributions to the pot. You therefore want to make the pot as big as possible: you should raise!

16 Bluffing A bluff is a bet or a raise in a situation where you are almost certain that you do not have the best hand. Does mathematics have a role to play in bluffing strategy?

17 A Proposition Wager We each ante $10. You take a ball from the bag, and either BET $10 or CHECK. If you bet, I can either CALL $10 or FOLD. In the event of a showdown: You Win if the number is 1-12 and I Win if the number is In addition, you have to tell me what your bluffing strategy is! We will play the game 30 times. Do you accept?

18 Optimal Bluffing Strategy Suppose you bluff by choosing some additional numbers in {13,14,,30} with which to bet. How many should you choose? A. 1 B. 2 C. 3 D. 4 E. 5

19 Explanation Suppose you bluff with exactly 4 additional numbers. Then I have two counter-strategies: Fold to any bet. Over 30 hands I win $10 each of the 14 times you do not bet, and lose $10 each of the 16 times you do. I therefore lose $20 every 30 hands. Call any bet. Again I win $10 each of the 14 times you do not bet. I will also win $20 each of the 4 times you bluff. But I will lose $20 each of the 12 times you bet with a winning number. My net profit is $140+$80-$240=-$20, so I lose $20 again. No matter which strategy I choose, I lose $20!