Introduction to Casino Management

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Mathematics

Casino Management

Temple University

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I need both Assignment's done on seperate files. Just include answers in the right floder. I gave you a nice powerpoint to help you with the question's

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Assignment Instructions & Requirements: Powerball (Links to an external site.)Links to an external site.is a lottery game which is a combined large jackpot game and a cash game. Every Wednesday and Saturday night at 10:59pm Eastern Time, a drawing of five white balls out of a drum with 69 balls and one red ball out of a drum with 26 red balls is conducted. Players win a prize by matching one of the 9 Ways to Win (Links to an external site.)Links to an external site.. The jackpot - won by matching all five white balls in any order and the red Powerball - is either an annuitized prize paid out over 29 years (30 payments counting the first immediate payment) or a cash lump sum payment. The Powerball estimated jackpot is currently $109,000,000. That is payable as an annuity over 30 years or one can receive the net present value of the annuity in a cash payment ($71,800,000). 1.) What is the jackpot’s expected value to the player who buys a $2 ticket for this drawing? Please show the expected value for both the jackpot as an annuity and for the net present value in cash. For this problem, ignore the other ways to win and focus only on the jackpot. Please show your work. Assignment Instructions & Requirements: Notes: • • • Any mention of a deck of cards refers to a standard 52 card deck, composed of four suits (♠ ♣ ♥ ♦) where (♠ ♣) are black and (♥ ♦) are red. Within each suit cards are ordered 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A, with J, Q, K, A considered face cards. Problems involving dice refer to standard 6 sided die and that we observe the face of the die facing upwards. All coins tossed are fair coins with two sides (heads and tails). Answer on a separate, typed document. Please show all necessary work in answering the problems. 1. From a shuffled deck of cards, what is the probability of drawing a face card that is a ♣? What is the probability of drawing a face card or a ♣? 2. When rolling a pair of dice, what is the probability that they will show a total of 8? 3. What is the probability of flipping a coin and observing three consecutive heads? 4. From a single deck of cards, assume that the following cards have already been played: J♦, 4♦, 10♥, A♥, 8♣, 5♣, 3♣, 7♠, 9♠. What is the probably of drawing a 4 or less on the next card? What is the probability of drawing a face card on the next card? 5. When rolling a pair of dice, what is the probability of rolling at least one seven in four rolls? Gambling Math GAMING & CASINO MANAGEMENT Gambling Math • How are Card Games like Blackjack different from games such as Roulette, Craps & Slots? • Roulette, craps and slots have elements of independence and randomness throughout • Card games odds are constantly changing due cards drawn • IE – cards drawn in the first hand will affect odds on subsequent hands Independent vs Dependent • Independent • Coin Flips • Dice • Roulette • Future events not based on past/current outcomes • Dependent • Cards • Previous hands of cards • impact future hands Once a card is dealt, it can never be dealt again Gambling Math Blackjack • Keeping track of which cards have been played can give the player clues as to which cards may come next • This does not guarantee success, but allows the player to utilize strategy Roulette or Craps • The wheel and the dice have no memory; keeping track of previous rounds provides no advantage Roulette Winners Casino Games • When the player enters the casino, she is faced with options: • Inter-Game Options • • What Game to play (ie – slots, blackjack, poker, roulette) Intra-Game Options • What game or table to play • In game options for betting (ie – win, place, show bets for horses) • Understanding the odds gives the player a better chance and possibly positive expected values Probability Gambling Math Quick Facts on Probability • If something cannot happen (impossible), it has a P = 0 • If something must happen (certainty), it has a P = 1 • Probability of occurrence X is referred to as the P(X) HEADS TAILS Quick Facts on Probability: Coin Toss • A Coin Flip: • 50% Chance of Heads • 50% Chance of Tails • Probability of Heads = • P(Heads) = 1 2 = .5 Heads Outcomes Total Outcomes Quick Facts on Probability: Dice • A Single Die has 6 sides, numbered 1,2,3,4,5,6 • Probability of any number is 1/6 or .167 P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = .167 .167 .167 .167 .167 .167 1.000 Games are played with Two Dice, not a single die, so the probability is based on: 6 X 6 = 36 Outcomes Dice Probability Examples • What is the Probability of getting a 7 when rolling two Dice? • For this, we need to know: P(7) = Number of Ways to Get 7 Total Outcomes (1,6) (2,5) (3,4) (4,3) (5,2) (6,1) P(7) = 36 6 = 36 1 = 6 = .167 Two Dice Outcomes & Probabilities Outcome X/36 Probability Roll a 2: (1,1) 1/36 0.0277778 Roll a 3: (1,2) , (2,1) 2/36 0.0555555 Roll a 4: (1,3), (2,2), (3,1) 3/36 0.0833333 Roll a 5: (1,4), (2,3), (3,2), (4,1) 4/36 0.1111111 Roll a 6: (1,5), (2,4), (3,3), (4,2), (5,1) 5/36 0.1388889 Roll a 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) 6/36 0.1666667 Roll a 8: (2,6), (3,5), (4,4), (5,3), (6,2) 5/36 0.1388889 Roll a 9: (3,6), (4,5), (5,4), (6,3) 4/36 0.1111111 Roll a 10: (4,6), (5,5), (6,4) 3/36 0.0833333 Roll a 11: (5,6), (6,5) 2/36 0.0555555 Roll a 12: (6,6) 1/36 0.0277778 Quick Facts on Probability: Cards • A Standard Deck of Cards: • 52 Cards, assumed random, shuffled • Four Suits: Hearts, Diamonds, Clubs, Spades • Aces, 2-10, Jack, Queen, King STANDARD DECK OF 52 CARDS Quick Facts on Probability: Cards • Typical Probabilities • Probability of any one card 1/52 = .019 • Probability of getting an Ace = 4/52 = 1/13 = .077 • Probability of getting a Spade = 13/52 = ¼ = .25 Expected Value Gambling Math Expected Value introduces Betting Odds to Probability If we know the probability of an outcome and the amount we wager and win in a bet, we can determine the Expected Value Expected Value E(X) = [P(W) x Amount Won] + [P(L) x Amount Wagered] Probability of Winning Dollar Amount of Gain in Win Probability of Losing Dollar Amount of Loss (THIS IS A NEGATIVE DOLLAR AMOUNT) Mathematical Expectation • The Mathematical Expectation for any game is as follows: • Multiply each possible gain or loss by the probability of that gain or loss • A “Fair Game” is one with an Expected Value of 0 • Neither the house nor player have an advantage Mathematical Expectation •An Example of Coin Toss • We know an unbiased coin, over time, will deliver 50% heads and 50% tails • If given a chance to bet even money ($1 bet = $1 win) on next toss • Mathematical Expectation in this example is: (.5)(1) + (.5)(-1) = 0 A Positive Expected Value Example • Suppose for our next coin toss, we’re offered 3/2 odds, that is a gain of $1.50 for our $1.00 bet • What is our formula? (.5)(1.50) + (.5)(-1) = +.25 • This game has a positive expected value of .25 • If we played this game 100 times, we’d expect to win $25 Blackjack Expected Value Example ODDS IN BLACKJACK • Object of Blackjack: Closest to 21 without going over • First hand of a single deck blackjack game. Deck is assumed to be shuffled and random. • Player Bets $100 and is dealt a 10, K (equaling 20) • The Dealer shows a 10 • In the deck there are now 49 cards including • 13 Tens (K, Q, J, 10) and • 32non-tens (2, 3, 4, 5, 6, 7, 8, 9) • 4 Aces • What are the odds that the Dealer wins? ODDS IN BLACKJACK • What are the odds that the Dealer wins? • First, if we have 10+10 and the Dealer has 10 showing, what card does the Dealer need to beat us? ODDS IN BLACKJACK • What are the odds that the Dealer wins? • The Dealer Needs an Ace • Ace has a Value of 11 • 11 + 10 = 21 > 10 + 10 = 20 • So, we want to find the probability of the Dealer holding an ACE ODDS IN BLACKJACK • So, we want to find the probability of the Dealer holding an ACE • We know there are three Tens showing, so in the decks there are now 49 cards including • 13 Tens (K, Q, J, 10) and • 32 non-ten numbered cards (2, 3, 4, 5, 6, 7, 8, 9) • 4 Aces CARD VALUES IN BLACKJACK 1 or 11 Face Value of the Card i.e. 2,3,4,5,6,7,8,9 10 A Note on ACE cards in Blackjack: An ACE can either have a value of 1 or 11, based on the needs of the player. In the following examples, we will treat the ACE as 11. ODDS IN BLACKJACK • So, we want to find the probability of the Dealer holding an Ace: P(ACE) = Number of Ways to Get Ace Total Outcomes = 4 49 = .0816 = .9184 • And the dealer not holding an Ace: P(NOT) = Number of Ways Not Ace Total Outcomes = 45 49 ODDS IN BLACKJACK • So, we want to find the probability of the Dealer holding an Ace: P(ACE) = Number of Ways to Get Ace Total Outcomes = 4 49 = .0816 Notice Adding these = 1 • And the dealer not holding an Ace: P(NOT) = Number of Ways Not Ace Total Outcomes = 45 49 = .9184 ODDS IN BLACKJACK • Now that we know the P(ACE) and P(NOT), we can determine the Expected Value of this Bet: E(X) = (.9184)($100) + (.0816)(-$100) = E(X) = ($91.84) + (-$8.16) = E(X)) = $83.68 The Bad News: This Bet doesn’t exist in Blackjack. You place your wager before you know what cards you’ll receive Let’s do a Real-World Example of Expected Value in Blackjack. The Insurance Bet EXPECTED VALUE IN BLACKJACK • First hand of a 4 deck blackjack game. Deck is shuffled and random • Player Bets $12 and is dealt a 6, 5 • In the 4 decks there are now 205 cards including • • 64 Tens (K, Q, J, 10) and 144 non-tens (2,3,4,5,6,7,8,9,A) • Dealers Shows ACE and offers insurance • Insurance is a $6 bet that pays $12 if Dealer has a 10 or $6 otherwise • The players EV on the Insurance Bet is: (64/205)(12) + (141/205)(-6) = -78/205 or -.3805 EXPECTED VALUE IN BLACKJACK CONTINUED (Tens/Remaining Cards)(Amount of Bet) + (Non-Tens/Remaining Cards)(Amount of Loss) (64/205)(12) + (141/205)(-6) = -78/205 (.3122)(12) + (.6878)(-6) = 3.7464 + -4.1268 = -0.3804 So what does this mean? -$.38 is a negative EV for the Player and thus the Player should decline insurance NOTE: Remaining Cards = 208 Total – (Player’s 2 cards, Dealer’s 1 Up Card) Expected Value of Dice A Craps Example EXPECTED VALUE OF A DICE GAME Similar to a Pass Line Bet in Craps, Joe is betting $10 that the roll of two dice does not equal a 2, 3 or 12 (32/36)($10) + (4/36)(-$10) (8.889) + (-1.111) = $7.778 Outcome X/36 Probability Roll a 2: (1,1) 1/36 0.0277778 Roll a 3: (1,2) , (2,1) 2/36 0.0555555 Roll a 4: (1,3), (2,2), (3,1) 3/36 0.0833333 Roll a 5: (1,4), (2,3), (3,2), (4,1) 4/36 0.1111111 Roll a 6: (1,5), (2,4), (3,3), (4,2), (5,1) 5/36 0.1388889 Roll a 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) 6/36 0.1666667 Roll a 8: (2,6), (3,5), (4,4), (5,3), (6,2) 5/36 0.1388889 Roll a 9: (3,6), (4,5), (5,4), (6,3) 4/36 0.1111111 Roll a 10: (4,6), (5,5), (6,4) 3/36 0.0833333 Roll a 11: (5,6), (6,5) 2/36 0.0555555 Roll a 12: (6,6) 1/36 0.0277778 EXPECTED VALUE OF A DICE GAME What if Joe were to bet “Don’t Pass” instead where he wins on a 2, 3 or 12? (4/36)($10) + (32/36)(-$10) (1.111) + (-8.889) = -$7.778 Outcome X/36 Probability Roll a 2: (1,1) 1/36 0.0277778 Roll a 3: (1,2) , (2,1) 2/36 0.0555555 Roll a 4: (1,3), (2,2), (3,1) 3/36 0.0833333 Roll a 5: (1,4), (2,3), (3,2), (4,1) 4/36 0.1111111 Roll a 6: (1,5), (2,4), (3,3), (4,2), (5,1) 5/36 0.1388889 Roll a 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) 6/36 0.1666667 Roll a 8: (2,6), (3,5), (4,4), (5,3), (6,2) 5/36 0.1388889 Roll a 9: (3,6), (4,5), (5,4), (6,3) 4/36 0.1111111 Roll a 10: (4,6), (5,5), (6,4) 3/36 0.0833333 Roll a 11: (5,6), (6,5) 2/36 0.0555555 Roll a 12: (6,6) 1/36 0.0277778 Expected Value of the Lottery EXPECTED VALUE OF A LOTTERY GAME Jim loves to play the lottery, but only when the jackpot is over $350,000,000, which is the mark it just hit today. Jim would elect take the cash payment, as opposed to the annuity payments, which would reduce the jackpot payout to $260,000,000. Jim is buying his $1 ticket, but is curious about the EV of this game (jackpot only). (1/258,890,850)($260,000,000) + (258,890,849/258,890,850)(-1) = (.0000000038626)($260,000,000) + (.9999988373479)(-1) = (1.0032842379327) + (-.9999988373479) = .0042854005848 The expected value is $0.00 These are the chances of winning the New York Mega Millions from nylottery.ny.gov IMPORTANCE OF REPETITION • It’s important to note that EV is the amount you will tend to win or lose on average over the course of repeated trials • Randomness plays a critical role in individual trials • Even Money Wager; $100 bet per toss; 10 Tosses • Randomness at play in 10 Tosses; over 1,000 or more tosses we’d see the trend to be even because it’s a Fair Game
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Assignment Instructions & Requirements:
Powerball (Links to an external site.)Links to an external site.is a lottery game which is
a combined large jackpot game and a cash game. Every Wednesday and Saturday
night at 10:59pm Eastern Time, a drawing of five white balls out of a drum with 69 balls
and one red ball out of a drum with 26 red balls is conducted. Players win a prize by
matching one of the 9 Ways to Win (Links to an external site.)Links to an external site..
The jackpot - won by matching all five white balls in any order and the red Powerball - is
either an annuitized prize paid out over 29 years (30 payments counting the first
immediate payment) or a cash lump sum payment.
The Powerball estimated jackpot is currently $109,000,000. That is payable as an
annuity over 30 years or one can receive the net present value of the annuity in a cash
payment ($71,800,000).
1.) What is the jackpot’s expected value to the player who buys a $2 ticket for this
drawing? Please show the expected value for both the jackpot as an annuity and
for the net present value in cash.
For this problem, ignore the other ways to win and focus only on the jackpot. Please
show your work.

Solution
The total number of different outcomes in d...


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