Name: Partners: everyone else in the class GAMBLING Games of chance, such as those using dice and cards, oporate according to the laws of statistics: the most probable roll is the one to bet on, and the least probable hand wins the card game. (Equipment: 6-sided die, deck of 52 cards, m&m s) OBJECTIVES 1. To calculate the probabilities of simple and complex systems. 2. To experimentally verify these probabilities. 3. To have some fun. PRE-LAB (to be completed before coming to lab) Prior to coming to lab, read through this write-up and perform all the exercises labeled Pre-Lab. OVERVIEW Statistics is the mathematics of making the most, in spite of missing information. In the case of thermodynamic systems, it is impossible to keep track of all 10 23 particles individual behaviors, yet we can build a physically sound model of the whole system by treating the individuals statistically. Key to statistics is bridging the gap between microscopic and macroscopic views for example, under the microscope, you may be able to distinguish Joe-flea from Jane-flea, but to the naked eye, they re two, indistinguishable fleas. In the case that all microscopic states are equally probable, the probability of a macroscopic state is simply the fraction of all microscopic states that achieve the desired macroscopic state, i.e., the macro-state s multiplicity divided the total number of micro-states: ( ) ( state) P state =. all ( )
Probabilities of contingent situations (those logically connected by an AND) multiply, while probabilities of alternative situations (those logically connected by an OR) add. Dice For the dice portion of this experiment, the system consists of two individual, distinguishable particles (6-sided die). The system s micro-states are the specific pairs of individual dice rolls: for example, one rolls a 3 and the other rolls a 4, so the microstate is described by the pair (3,4). The macro-state is defined by the sum of the two die values, in this example, 3 + 4 = 7 is the macro-state s value. Since it would be impractical to model a dice s role (initial momentum, collisions with table, ), the best we can do is assume each of the 6 possible states is equally probable and treat the dice roles statistically. For an individual dice, there are six equally probable states, so 1 for example the probability of rolling a three is P ( 3) =. Similarly, that of 6 1 rolling a 4 is P ( 4) =. 6 Cards For the poker portion of this experiment, the particles are 52 cards. Since we don t know their ordering in a shuffled deck, the best we can do is treat the system statistically. For both systems, we ll compare the calculated probabilities (Pre-lab) of out comes with the number of actual incidents (in lab). PART ONE: Dice Pre-Lab: On a separate sheet, calculate the probabilities of each of the possible macro-states. You will turn in that separate sheet. To get you started, I ll do one: ( 7) ( all ) P( 7 ) =, ( 7) = 6 since there are 6 micro-states (rolls) that yield a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1); ( all ) = 36 for there are 6 possible rolls of the first dice, and for each of those there are 6 possible rolls of the second dice, giving 6 6 = 36 possible two-dice roles. Page 2
( 7) 6 1 = ( all ) 36 6 P ( 7 ) = =. Rules. Everyone will have a pair of dice, in one round, everyone roles their own dice. Who ever makes the least likely roll wins. In the event of a tie, the roll with the actual higher value wins, ex. a 12 is equally likely as a 2, but 12 >2 so the 12 wins. If there is a true tie (say, two players roll a 12), then the proceeds are split. 1. Play for about 10 minutes, and record each of your own roll s scores on a separate sheet (don t bother recording the individual die s values, just the sum, i.e. macro-states). Also note whether or not you won the round. 2. At the end of the game, collect the scores of your class mates. 3. Tabulate the number of incidents of each score (ex., how many times a 7 was rolled, how many times an 8 was rolled, ) and enter them in the table below. 4. Divide these numbers of incidents by the total number of rolls to get the fraction of incidents, enter these values in the table below. 5. Comparing the probability and fraction of incidents values, what is percent deviation? Enter these values in the table below. Macro Prob. State Theory 1 2 3 4 5 6 7 1/6 8 9 10 11 12 # of Incidents Fraction of incidents % deviation Page 3
6. Do you see any trend in the deviations? Can you explain it? 7. Assuming a fair game (that each player was equally likely to win each round), what is the % deviation of the fraction of your winnings from your expected fraction? Can you explain it? Page 4
PART TWO: Poker Pre-Lab: On a separate sheet, calculate the probabilities that an individual player is dealt the hands listed in the table below. This list is in order from highest to lowest rank (lowest to highest probability). You will turn in that separate sheet. Fill in the probabilities in the table below. Note: there are plenty of places to look up the probabilities, and those are fine for checking yourself, but you need to work them out yourself. To get you started I ll do one. Full-House. A full house is one triplet AND one pair. How many ways of choosing the kind of the triplet? There are 13 kinds (A,2,3,4,5,6,7,8,9,10, J,Q,K), so the kind of the 3 matching cards has 13 possibilities. AND How many ways of choosing the suits of the 3? There are 4 suits and each of the 3 will necessarily be of different suits, so 4 how many ways are there to choose 3 of 4 suits: = 4 3 AND Now for the pair, how many ways of choosing its kind? It can be of any of the remaining 12 kinds. AND How many ways of choosing their kinds? Again any of the 4 suits, but now we re dealing with a pair, so we re choosing only 4 2 cards: = 6. 2 Now, the number of ways of getting a full house is the product of the number of ways of making each independent choice (the product of all the bold-faced numbers): 13*4*12*6= 3,744. That may seem like an awful lot! But of course, the probability of getting a full house depends on the ratio of the ways of getting it to the ways of getting any hand. Page 5
As for ( all. hands), it s a question of how many ways to select a hand of 5 cards from deck of 52 cards: 52 ( all. hands) = = 2,598,960. 5 So ( full. house) ( ull. house) ( all. hands) 3,744 2,598,960 3 P = = = 1.440576 10. Not so terribly likely! Now you do the rest. Pre-Lab. On a separate piece of paper, qualitatively, what will happen to the calculated probabilities if a combined deck of 104 cards (two decks) were to be used? Quantitatively, support this by calculating the probability of a Royal Flush when two decks are used. Rules. 5 card poker with no wilds. We unfortunately don t have time for bluffing, so cards are dealt face up. The pecking-order is that shown in the table below. In the event that there is a tie, the general rule is that the hand with the highest value feature cards wins, ex. pair of jacks beats a pair of nines. If that doesn t resolve it, one looks to the un-featured cards, ex. a pair of jacks with an ace beats a pair of jacks with an eight. If it s still a tie, one compares suits, let s say heart beats diamond beats spade beats club. 1. Until the last 10 minutes of class, we ll deal out rounds of hands, collecting and re-shuffling between them. Keep track of the types of hands you get, ex. a flush, pair,... (Don t bother recording the individual card values). Meanwhile, keep track of the rounds you won. 2. When the game is through, compile the list of hands of your class mates. 3. Tabulate the number of incidents of each hand and enter them in the table below. 4. Divide these numbers of incidents by the total number of hands dealt (one round with six players would count as a total of six hands, not just one.) to get the fraction of incidents; enter these values in the table below. 5. Comparing the probability and fraction of incidents values, what is percent deviation? Enter these values in the table below. Page 6
Macro State Prob. Theory # of Incidents Prop. Exp. % deviation Royal Flush 1 Straight Flush 2 Four of a Kind 2.40096 10-4 Full House 3 Flush 4 Straight 5 Three of a Kind Two Pairs One Pair Nothing 6 6. Do you see any trend in the deviations? Can you explain it? 7. Assuming a fair game (that each player was equally likely to win each round), what is the % deviation of the fraction of your winnings from your expected fraction? Can you explain it? 1 Ace, King, Queen, Jack, 10 all of the same suite. 2 Non-Royal. five contiguous numbers, all of the same suit. It may be easiest to calculate the probability for any kind of Straight Flush, and then subtract from it the probability of a Royal Flush. 3 Three of one kind, two of another. 4 Non-Straight. It may be easiest to calculate the probability for any Flush and then subtract the probabilities of Straight and Royal Flushes from it. 5 Non-Flush, five contiguous numbers. It may be easiest to calculate the probability of any straight and subtract the probabilities of Royal Flush and Straight Flush from it. 6 It may be easiest to simply subtract all the above probabilities from that of getting any hand, 1. Page 7