HMD 436 Name_______________________

Test #1

March 4, 2002

For full credit show your work and the numeric answer. A fraction, decimal, or percent are all acceptable. All questions are worth 4 points.

1-4: Consider the bet on 11 in craps. The bet pays 15 to 1 if the next roll is 11, otherwise the bet loses.
  1. What are the payoff odds?

    2. 15-1
  2. What is the probability of winning?

    3. 2/36
  3. What are the true odds?

    4. There are 2 ways to win and 34 ways to lose. So the true odds are 34-2 = 17-1.
  4. What is the house edge?

The expected value is (2/36)*15 + (34/36)*-1 = -4/36 = -11.11%. The house edge is –1*expected value, which equals 11.11%.

You can also use this formula (true odds-payoff odd)/(true odds+1) = (17-15)/(17+1) = 2/18 = 1/9 = 11.11%.

5. In Wild Hold ‘em Fold ‘em the player’s expected loss per initial wager is 6.86%. The player has the option to raise after his initial bet. The average total bet is 2.13 times the initial wager. What is the element of risk?

6.86%/2.13 = 3.22%

6. In Three Card Poker there are two bets to choose from, Pairplus and Ante & Play. The following table shows the house edge and element of risk of each bet.

Game

House Edge

Element of Risk

Pairplus

2.32%

2.32%

Ante & Play

3.37%

2.01%

  2. If the player wishes to minimize the ratio of expected losses to total amount bet, which bet should he make and what would that ratio be?

The player should play the bet with the lower element of risk, which is Ante & Play, at 2.01%.

7-9: Players purchase $24,000 in chips over an 8-hour shift at a blackjack table. The players take $21,000 in chips from the table to cash out.

  1. What is the drop?

    2. $24,000
  2. What is the win?

    3. $24,000-$21,000=$3000
  3. What is the hold percentage?

$3000/$24000=3/24=1/8=12.5%.

10. A player roulette player bets $5 at a time for 4 hours. Assume the dealer’s speed is 50 hands per hour. The house edge in roulette is 5.26%. What is the player’s handle and expected loss? (2 points each)

Handle=$5*50*4=$1000

Expected loss = $1000*0.0526 = $50.26

 

11. The Bellagio offers single zero roulette. If the player makes an even money bet (like red or black) and the ball lands in zero the player will only lose half his bet. What is the expected value on the even money bets?

(18/37)*1 + (18/37)*-1 + (1/37)*-0.5 = -0.5/37 = 1.35%.

 

12. You have an ordinary deck of cards with one card missing. The missing card is a spade. You draw 4 cards out of the 51 at random. What is the probability they are all the same suit?

(3*combin(13,4)+combin(12,4))/combin(51,4) = (3*715+495)/249900 = 2640/249900 = 0.0106.

 

13-14: Consider the 15 bet in Sic Bo. If the player rolls a total of 15 with 3 dice the bet pays 14 to 1.
  1. What is the probability of rolling a 15?

6+6+3 (3 ways)

6+5+4 (6 ways)

5+5+5 (1 way)

Total = 10 ways.

Total combinations = 6^3=216

Probability = 10/216 = 4.63%.

14. What is the player’s expected value?

(10/216)*17 + (206/216)*-1 = -36/216 = -16.67%.

 

15-16: The Big Eight bet in craps pays 1 to 1 (even money) if the player rolls an 8 before a 7.
  1. What is the probability of winning?

    2. Probability of 7 = 6/36

    Probability of 8 = 5/36

    Probability if 8 before 7 = (5/36)/(6/36+5/36) = 5/11 = 45.45%.
  2. What is the player’s expected value?

(5/11)*1 + (6/11)*-1 = -1/11 = -9.09%.

 

17-23: Consider is modified Uno deck with four colors (red, blue, yellow, and green) numbered 1 to 10 each. There is one of each card for a total of 40 cards. The lowest straight is 1-2-3-4-5 and the highest is 6-7-8-9-10. The 10 can only be counted as high. There is no royal flush (colored 6-7-8-9-10 counts as a straight flush). Indicate the total number of possible combinations for each hand.
  1. Straight flush (all the same color in order)

6*4=24

18. Four of a kind

10*9*4=360

19. Full house

10*9*combin(4,3)*combin(4,2)=10*9*4*6=2160

 

20. Flush (all the same color)

4*(combin(10,5)-6) = 4*(252-6) = 984

  1. Straight (in order)

    2. 6*(4^5-4) = 6*(1024-4) = 6120
  2. Three of a kind

    3. 10*combin(9,2)*combin(4,3)*combin(4,1)^2=10*36*4*4*4=23,040
  3. Two pair

    4. Combin(10,2)*8*combin(4,2)^2*combin(4,1)=45*8*36*4=51840
  4. On The Price is Right there is a game called the Race Game in which the player must match four different price tags to four different prizes. Assuming the player is randomly guessing what is the probability of winning on the first try?

There are 4!=24 possible ways to order the four price tags. So the probability is 1/24=4.17%.

25. On Who Wants to Be a Millionaire the contestant is on the last question. If he gets it right he wins $1,000,000. If he gets it wrong he wins $32,000. The player also has the option to not guess and walk away with $500,000. The player’s probability of getting the question right is p. At what value of p is the expected value the same for guessing and walking?

p*1000000 + (1-p)*32000 = 500000

968000*p + 32000 = 500000

968000*p = 468000

p = 468000/968000 = 48.35%.