HMD 436 Name___________________

Final Exam

May 13, 2002 6:00-800 PM

 

  1. What is the probability of rolling a total of 2 with two ordinary dice? (2 points)

    2.  

    (1/6)^2 = 1/36

     

     

  2. What is the probability of rolling a total of 6 with two ordinary dice? (3 points)

    3. Ways to roll a 6: 1+5, 2+4, 3+3, 4+2, 5+1 = 5. So the probability is 5/6^2 = 5/36

     

  3. Two cards are drawn from an ordinary deck without replacement. What is the probability they are both the same rank? (3 points)

    4. 13*combin(4,2)/combin(52,2) = 13*6/1326 = 1/17 = 0.0588

     

  4. What is the probability of drawing a blackjack from a 4-deck shoe? (3 points)

    5.  

    (4*4)*(16*4)/combin(52*4,2) = 16*64/21528 = 0.0476

     

     

     

  5. You have seven rooms in your house. Every day you clean each room. To avoid boredom you clean then in a different order every day. How many days can you go before repeating an order? (3 points)

    6. 7! = 5040

     

  6. At your favorite Las Vegas arcade there are 12 video games. Each day you play a different combination of four of them. How many days can you go without playing the same combination twice? (3 points)

    7. Combin(12,4) = 495

     

  7. An urn contains 10 black balls and 15 white balls. You draw 6 balls at random without replacement. What is the probability you draw 2 black balls and 4 white balls? (3 points)

    8.  

    Combin(10,2)*combin(15,4)/combin(25,6) = 45*1365/177100 = 0.3468

  8. Two ordinary dice are rolled until a total of 5 is obtained. What is the expected number of rolls this will take? (3 points)

    9. Ways to roll a 5: 1+4, 2+3, 3+2, 4+1 = 4.

    Probability of a 5: 4/36

    Expected number of rolls is 1/(4/36) = 36/4 = 9

     

     

  9. How many different combinations of 7 cards can be dealt from a single deck of cards? (3 points)

    10.  

    Combin(52,7) = 133784560

     

  10. What is the probability of drawing a full house from a single deck of ordinary cards? (3 points)

    11.  

    13*12*combin(4,3)*combin(4,2)/combin(52,5) = 13*12*4*6/2598960 = 3744/2598960 = 0.0014406.

     

  11. The shooter in craps must roll a total of 9 before rolling a 7. He keeps rolling until either total is obtained. What is the probability of rolling the 9 first? (3 points)

Probability of 7 = 6/36

Probability of 9 = 4/36

Probability of 9 before 7 = (4/36)/((6/36)+(4/36)) = (4/36)/(10/36) = 4/10 = 2/5

 

 

12. What is the probability of being dealt (before the draw) a sequential royal flush in either direction in normal 52-card deck video poker? (4 points)

 

2*4/permut(52,5) = 8/311875200=2.5651*10^-8

For questions 13,14 consider a modified deck with 16 ranks and 6 suits. The ace can be high or low.

  1. How many combinations are there for a flush (not including straight flush)? (5 points)

    2.  

    6*(combin(16,5)-13) = 6*(4368-13) = 26130.

  2. How many combinations are there for a two pair? (5 points)

    3.  

    Combin(16,2)*14*combin(6,2)^2*6 = 120*14*15^2*6 = 2268000

    Problems 15,16 are all based on blackjack side bets.

  3. In an eight-deck shoe what is the probability of drawing four aces of the same color as the first four cards? (5 points)

    4. 2*combin(2*8,4)/combin(8*52,4) = 2*combin(16,4)/combin(416,4) = 2*1820/1229930520 = 2.9595*10^-6

     

     

  4. In a six-deck shoe what is the probability of drawing two non-suited sevens as the first two cards, and then a non-7 as the third card? (5 points)

    5.  

    permut(4,2)*6^2*(48*6)/permut(52*6,3) = 12*6^2*288/30079920 = 0.0041362

     

     

     

  5. In bingo what is the probability the player will form a frame shape (covering all the edges of the card) within 50 calls? (6 points)

    6. Total numbers along frame = 16.

    Total numbers not on frame = 75-16 = 59

    Total calls other than frame numbers = 50-16 = 34

    Combin(59,34)/combin(75,50) = (3.0284*10^16)/(5.2589*10^19) = 0.00057587

     

     

     

  6. What is the probability in a room of 500 cards at least one player will form a frame shape within 50 calls? (4 points)

    7. 1-(1-0.00057587)^500 = 0.250251

  7. Consider the following money line.

    8. Chicago (-140) Vs. Houston (+120).

    Assuming a fair set of lines would have been -130 and +130 what is the probability that Atlanta will win? (4 points)

    P*100 + (1-p)*-130 = 0

    230p = 130

    p=130/230 = 13/23 = 0.5652

     

     

  8. From the above problem what is the expected value for a bet on Chicago? (4 points)

    9. ((13/23)*100 + (10/23)*-140)/140 = (-100/23)/140 = -0.0311

     

     

  9. Assume the player has a 70% chance of correctly picking a football game with a free 7 points. A 5-team 7-point teaser pays 7-2. What is the expected value? Ignore ties. (5 points)

    10.  

    Probability of win = .7^5 = 0.1681

    Probability of loss = 1-.7^5 = 0.8319

    EV = .1681*(7/2) + .8319*-1 = -0.2437

     

     

     

  10. The total amount bet on place bets at Prairie Meadows is $100,000. The track cut on place bets is 18%. The two placing horses are 4 and 7. $6000 is bet on 4 and $9000 on 9. How much is a $20 place bet on horse 4 worth (including the original $20 which is refunded)? The track has a 10-cent breakage. (5 points)

    11. Original pool = $100,000

    After track cut = $100,000*82% = $82,000

    After winning bets = $82,000-$6000-$9000 = $67,000

    Each place bet pool = $67,000/2 = $33,500

    Ratio of winnings to original bet = $33,500/$6000 = 5.5833

    Value of $2 ticket = $2+$2*(5.5833) = $2 + $11.17 = $13.17

    Value of $2 ticket after rounding = $13.10

    Value of $20 ticket = $22.30 * ($20/$2) = $131.00

     

     

     

     

     

  11. A double-zero roulette player bets $1 on a 4-number combination bet (which pays 8-1) 500 times. Using the normal distribution is what is the probability that the player wins $100 or more? Also indicate the Z statistic. (8 points)

    12.  

    Event

    Probability

    Pays

    E(x)

    E(x^2)

    Win

    0.105263

    8

    0.842105

    6.736842

    Loss

    0.894737

    -1

    -0.894737

    0.894737

    Total

    1.000000

    -0.052632

    7.631579

    Variance per spin = 7.681579 – (-0.052632)^2 = 7.628809

    Standard deviation = 7.628809^0.5 = 2.762030

    Total standard deviation = 2.762030*500^0.5 = 61.760865

    Expected win = 500*(-0.052632) = -26.315789

    Z = (100-0.5-(-26.315789))/ 61.760865 = 2.037144

    Pr (z>=2.037144) = 1-pr(z<=2.037144) = 0.020818

     

  12. A casino uses a promotional prize wheel. The wheel should stop on each prize with the following probabilities:

    13. Key chain 30%

    Deck of cards 20%

    Buffet coupon 15%

    Show ticket 10%

    Dice clock 10%

    T-shirt 10%

    Jacket 4%

    $1000 1%

    Following is the wheel’s actual and expected results over 10,000 spins. You may use the chi-squared column for your own work. What is the chi-squared statistic and what is the probability that a fair test would be this skewed or more? (8 points)

    Prize

    Expected

    Actual

    Chi-squared

    Key chain

    3000

    2970

    0.3

    Deck of cards

    2000

    1920

    3.2

    Buffet

    1500

    1440

    2.4

    Show ticket

    1000

    1020

    0.4

    Dice clock

    1000

    1040

    1.6

    T-shirt

    1000

    1060

    3.6

    Jacket

    400

    430

    2.25

    $1,000

    100

    120

    4

    Total

    10000

    10000

    17.75

     

    Probability that chi-squared statistic with 7 degrees of freedom is greater than 17.75 = chidist(17.75,7) = 0.013150336

     

     

     

  13. Briefly explain how Japanese pachinko parlors get around the law prohibiting the exchange of balls back into cash? (4 bonus points)

Pachinko balls can be redeemed for prizes in the pachinko parlor. Then the player takes the prizes to a nearby building and sells them for cash.