HMD 436 Test #2 Name______________________

April 8, 2002

 

Consider a three reel single-line slot machine with weighted reels. The following table shows the total weights per reel for each symbol.

Symbol

Reel 1

Reel 2

Reel 3

Bar

3

2

1

Bell

4

3

2

Plum

5

4

3

Orange

6

5

4

Blank

14

18

22

Total

32

32

32

1. 3 bars pays 1200, 3 bells pays 400, 3 plums pays 100, and 3 oranges pays 75, all per coin bet. Use the following table if you wish or Excel to determine the player’s expected return. (15 points total)

Payline

Combinations

Pays

Return

3 bars

6

1200

7200

3 bells

24

400

9600

3 plums

60

100

6000

3 oranges

120

75

9000

Total

210

31800

 

 

 

Total return = ___31800______________

 

 

32^3=32768

Total combinations = __32768________________

 

31800/32768 = 0.9705

 

Expected return = __0.9705__________________

 

2. Consider a modified keno game in which there are 60 balls in the chamber, of which 15 will be drawn at random. The pick 6 game pays 1 for matching 3, 10 for matching 4, 100 for matching 5, and 1000 for matching 6. Use the following table if you wish or Excel for your work. (20 points total)

Match

Combinations

Pays

Return

3

6,860,137,775,400

1

6,860,137,775,400

4

1,435,842,790,200

10

14,358,427,902,000

5

143,584,279,020

100

14,358,427,902,000

6

5,317,936,260

1000

5,317,936,260,000

Total

8,444,882,780,880

40,894,929,839,400

 

 

 

 

 

 

 

Total return = __40,894,929,839,400_________________

 

 

Combin(60,15)= 53,194,089,192,720

Total combinations = __53,194,089,192,720__________________

 

40,894,929,839,400/53,194,089,192,720=0.768787105

 

Expected return = ___0.768787105__________________

 

 

3. Consider a game of Derby with the following pay table. You may use the following table or Excel for your own work. (15 points total)

Quinella

Pays

Fair probability

True probability

Return

1,2

95

0.010526316

0.009158708

0.870077

1,3

4

0.25

1,4

9

0.111111111

1,5

37

0.027027027

2,3

21

0.047619048

2,4

49

0.020408163

2,5

200

0.005

3,4

2

0.5

3,5

8

0.125

4,5

19

0.052631579

Total

1.149323244

 

 

Fair probability of 1,2 quinella ____0.010526316__________________

 

 

 

 

Sum of all ten fair probabilities ____1.149323244___________________

 

0.010526316/1.149323244 = 0.009158708

Actual probability of 1,2 win _____0.009158708__________________

 

95 * 0.009158708 = 0.870077, or…

1/1.149323244 = 0.870077

Expected return of 1,2 win __0.870077______________________

 

4. Consider the Royal Match side bet played with 5 decks. (20 points total)

How many combinations are there for a "royal match", in other words a suited king and queen?

4*5^2=4*25=100

 

How many combinations are there for a "easy match", in other words any suited pair besides a royal match?

4*((combin(13,2)-1)*5^2 + 13*combin(5,2)) = 4*(77*25 + 13*10) = 8220

 

How many combinations are there for two non-suited cards?

Combin(4,2)*(13*5)^2 = 6*4225 = 25350

 

Use the following table or Excel to determine the total return.

Hand

Combinations

Pays

Value

Royal match

100

25

2500

Easy match

8220

2.5

20550

Non-suited

25350

-1

-25350

Total

33670

-2300

Total value = _____-2300_______________

 

 

 

Total combinations = ____33670________________

 

-2300/33670=

Player’s expected value = ___-0.0683____________________

 

 

5. What is the probability of 2 suited sevens as the first two cards and then a non-seven in a 7-deck shoe of normal 52-card decks? (5 points)

(4*permut(7,2)*(48*7))/permut(52*7,3) = (4*42*336) / 47831784 = 0.001180

 

  1. What is the probability the player will get two queen of hearts in his first two cards and then the dealer gets a blackjack in a 4-deck game? (5 points)

    2. (permut(4,2)*permut(2,2)*(4*4)*(16*4-2))/permut(52*4,4) = (12*2*16*62)/1818254880 = 0.00001309

     

  2. Consider a game of Pairplus in Three Card Poker with a modified 75-card deck of 15 ranks and 5 suits. (15 points total)

    3.  

    (15-1)*(5^3-5) = 14*120 = 1680

    How many combinations are there for a straight (not including a straight flush)?_____1680______________

     

     

    5*(combin(15,3)-(15-1)) = 5*(455-14) = 2205

    How many combinations are there for a flush (not including a straight flush)?____2205_______________

     

    15*14*combin(5,2)*5 = 10500

    How many combinations are there for a pair?___10500________________

     

  3. Consider the same modified deck as in problem 7. The casino uses 8 such decks in Casino War. What is the probability that a hand will result in a double tie (the first two cards tie and then the second two cards tie)? (5 points)

(15*permut(5*8,2)*(14*permut(5*8,2)+permut(5*8-2,2)))/permut(15*5*8,4) = 0.004239

Or….

Probability of first tie = 15*COMBIN(5*8,2)/COMBIN(75*8,2) = 0.065109

Probability of second tie = (14*COMBIN(5*8,2)+COMBIN(40-2,2))/COMBIN(75*8-2,2) = 0.065114

Probability of two consecutive ties = 0.065109 * 0.065114 = 0.004239