GAM 470 Test #2

April 5, 2005

Name_______Answers___________

 

Consider a keno game in which the casino draws 10 balls out of 40. A bet is available on a pick 6.

  1. How many ways can the player catch 3 out of 6?

    2. Combin(10,3)*combin(30,3) = 120*4060 = 487200

     

     

  2. How many ways can the player catch 4 out of 6?

    3. Combin(10,4)*combin(30,2) = 210*435 = 91,350

     

     

  3. How many ways can the player catch 5 out of 6?

    4. Combin(10,5)*30 = 252*30 = 7560

     

     

  4. How many ways can the player catch 6 out of 6?

Combin(10,6) = 210

 

 

 

5. What is the total number of combinations for picking 6 numbers out of 40?

Combin(40,6) = 3838380

 

 

 

6. Following is the pay table. Fill in the rest of the table to find the total win over all combinations.

Catch

Pays

Probability Combinations

Return Combinations

0

0

1

0

2

0

3

3

487200

1461600

4

10

91350

913500

5

100

7560

756000

6

2000

210

420000

Total

.

3838380

3551100

What is the expected return?

3551100/3838380 = 92.52%.

 

7. At an Internet casino you see the following bets available on who will win the 2008 presidential election. Assume the expected return is the same on all bets, what is it?

Bet

Pays

Arnold Schwarzenegger

3 to 1

Jeb Bush

5 to 1

John McCain

7 to 1

Hillary Clinton

2 to 1

John Edwards

8 to 1

Rudy Guiliani

4 to 1

Field

15 to 1

Bet

Pays

Pays (for one)

Inverse

Arnold Schwarzenegger

3 to 1

4

0.250000

Jeb Bush

5 to 1

6

0.166667

John McCain

7 to 1

8

0.125000

Hillary Clinton

2 to 1

3

0.333333

John Edwards

8 to 1

9

0.111111

Rudy Guiliani

4 to 1

5

0.200000

Field

15 to 1

16

0.062500

Total

.

.

1.248611

As the above table shows the player must bet 1.248611 units to get back 1 unit. So the expected return is 1/1.248611 = 0.80089, or 80.09%.

 

  1. You question the fairness of a video poker game on the deal. So you test 649,740 hands. The following table shows your expected and actual results. Fill in the last row to find the chi-squared statistic.

Hand

Expected

Actual

Measure of Deviation

Royal flush

1

2

1.0000

Straight flush

9

7

0.4444

Four of a kind

156

175

2.3141

Full house

936

1000

4.3761

Flush

1277

1250

0.5709

Straight

2550

2450

3.9216

Three of a kind

13728

13800

0.3776

Two pair

30888

31000

0.4061

Jacks or better

84480

85000

3.2008

All other

515715

515056

0.8421

Total

649740

649740

17.4536

What is the chi-squared statistic?_______17.4536_____________

  1. What is the p value?______=chidist(17.4536,9) = 0.042068_________

    2. For problems 11 to 16 consider a blackjack side bet based on the player’s first two cards and the dealer’s up card. There are five decks in the shoe.

  2. How many ways can the player get 3 suited fives?

    3. 4*combin(5,3) = 40

     

     

     

  3. How many ways can the player get a suited 4, 5, 6 (one of each)?

    4. 4*5^3 = 500

     

     

  4. How many ways can the player get a non-suited 555?

    5. Combin(4*5,3) – 4*combin(5,3) = 1140 – 40 = 1100

     

     

     

  5. How many ways can the player get a non-suited 4, 5, 6 (one of each)?

    6. (4*5)^3 - 4*5^3 = 8000 – 500 = 7500

     

     

     

  6. How many ways can the player get a flush (not including suited 555 and 456)?

    7. 4*combin(13*5,3) – 40 – 500 = 4*43680 – 540 = 174,180

     

     

     

     

     

  7. Fill in the following table to find the expected return.

    8. Hand

    Pays

    Probability Combinations

    Return Combinations

    Suited 555

    5000 to 1

    40

    200000

    Suited 456

    500 to 1

    500

    250000

    Non-suited 555

    200 to 1

    1100

    220000

    Non-suited 456

    100 to 1

    7500

    750000

    Flush

    5 to 1

    174180

    870900

    All other

    Loss

    2712300

    -2712300

    Total

    .

    2895620

    -421400

    Expected return = ________-14.55%________________

     

  8. At the Cal-Neva in Reno a bet on a player blackjack pays 17 to 1. What is the house edge in a 2-deck game?

    9. Probability of blackjack = (2*4)*(2*16)/combin(2*52,2) = 0.047796863.

    Probability of losing = 1-0.047796863 = 0.952203

    Expected value = 17*0.047796863 - 0.952203 = -0.13966.

    So the house edge is +13.966%.

     

  9. Your sock drawer has 7 blue socks, 5 green socks, and 3 red socks. You pick 5 socks at random and without replacement. What is the probability you pick 3 blue socks and 2 green socks?

    10. 3 blue sock combinations = combin(7,3) = 35

    2 green sock combinations = combin(5,2) = 10

    Total successful combinations = 35*10 = 350.

    Total all combinations = combin(15,5) = 3003.

    Probability = 350/3003 = 11.66%.

  10. In the game of Clue® there are 9 room cards, 6 weapon cards, and 6 suspect cards. You shuffle them and draw 3 at random. What is the probability you draw one of each?

Successful combinations = 9*6*6 = 324.

Total all combinations = combin(21,3) = 1330.

Probability = 324/1330 = 24.36%.

19 You remove all the face cards from a deck of cards. You then draw 5 cards from the remaining 40. What is the probability you get a two pair? (6 points)

Successful combinations = combin(10,2)*8*combin(4,2)^2*4 = 45*8*6^2*4 = 51840

Total all combinations = combin(40,5) = 658,008

Probability = 51840/658008 = 7.88%.