Casinos rely on the laws of probability and expected values of random variables to guarentee them profits on a daily basis. Some individuals will walk away very wealthy, while others will leave with nothing but memories. This activity is designed to allow you to analyze some of the games of chance that are typically played in casinos.

**Station 1: CRAPS**

Roll a pair of six sided dice. If the sum is 7 or 11, you win. If the sum is 2, 3, or 12, you lose. If the sum is any other number, you roll again. In fact, you continue throwing the dice unti you either roll that number again (WIN!) or roll a 7 (LOSE!).

a. **Simulation I:** Play 20 games of craps
with your partner. Each of you should throw the dice for 10 games.
Record you results in the table below.

Game | 1st Roll | Result | Subsequent Result |

1 | |||

2 | |||

3 | |||

4 | |||

5 | |||

6 | |||

7 | |||

8 | |||

9 | |||

10 | |||

11 | |||

12 | |||

13 | |||

14 | |||

15 | |||

16 | |||

17 | |||

18 | |||

19 | |||

20 |

In what proportion of the games did you win on your first roll?

In what proportion of the games did you win?

b. **Simulation II:** Using your TI-83,
you can simulate rolling two dice and obtaining their sum by tryping:
RandInt(1,6) + RandInt(1,6) and pressing ENTER. Simulate 20 games,
10 each, using your calculator. Record your results in the table
below.

Game | 1st Roll | Result | Subsequent Result |

1 | |||

2 | |||

3 | |||

4 | |||

5 | |||

6 | |||

7 | |||

8 | |||

9 | |||

10 | |||

1 | |||

2 | |||

3 | |||

4 | |||

5 | |||

6 | |||

7 | |||

8 | |||

9 | |||

10 |

c. **Probability Questions**

1. What is the probability that you obtain a sum of 7 or a sum of 11 on the 1st roll?

2. What is the probability that you obtain a sum of 2, 3, or 12 on the 1st roll?

3. What is the probability that you roll again after the 1st roll?

4. Suppose you roll a sum of 8 on the 1st roll. Find the probability that you subsequently win the game, given that you rolled an 8 to start with.

**Station 2: BLACKJACK**

The game of blackjack begins by dealing 2 cards to a player, the first face-down and the second face-up on top of the first. The player has a "blackjack" if he has a black jack and an ace as his or her two cards. The player has "twenty-one" if he has an ace and a 10, Jack, Queen, or King.

1. Deal 10 blackjack hands, one at a time, shuffling between each hand. That is, deal 2 cards, then check the result, then shuffle, then deal two more cards, etc. Record the number of "blackjacks" and "twenty-ones" you obtain:

Blackjacks:

Twentyones:

2. Given that the face-up card is an ace, find the probability that you have:

a. a blackjack

b. twentyone

3. Given that the face-up card is a black jack, find the probability that you have:

a. a blackjack

b. twentyone

4. Find the probability of getting a "blackjack".

5. Are they events A = face-up card is a black jack and B = you get "blackjack"

a. independent?

b. disjoint?