	4)	A survey of 160 people at a casino was conducted to determine the casino games that they typically play.  The following data was collected.  Complete the Venn Diagram.  (Label each of the circles.)
	
99 people played blackjack
77 people played roulette
39 people played craps
51 people played blackjack and roulette
22 people played blackjack and craps
11 played roulette and craps
9 people played blackjack, roulette, and craps


To solve this problem, we need to fill out a Venn diagram representing the number of people who played blackjack (B), roulette (R), and craps (C) based on the data provided. We will calculate the values for the intersections of the three sets using a step-by-step approach.

Here’s the breakdown of the information:

- 99 people played blackjack.
- 77 people played roulette.
- 39 people played craps.
- 51 people played blackjack and roulette.
- 22 people played blackjack and craps.
- 11 people played roulette and craps.
- 9 people played blackjack, roulette, and craps.

### Step-by-Step Calculations:

1. **People who played all three games (B ∩ R ∩ C):**
   - 9 people played blackjack, roulette, and craps.

2. **People who played blackjack and roulette, but not craps (B ∩ R):**
   - 51 people played blackjack and roulette, but this includes the 9 who played all three games. So, the number of people who played only blackjack and roulette is:
     \[
     51 - 9 = 42
     \]

3. **People who played blackjack and craps, but not roulette (B ∩ C):**
   - 22 people played blackjack and craps, but this includes the 9 who played all three games. So, the number of people who played only blackjack and craps is:
     \[
     22 - 9 = 13
     \]

4. **People who played roulette and craps, but not blackjack (R ∩ C):**
   - 11 people played roulette and craps, but this includes the 9 who played all three games. So, the number of people who played only roulette and craps is:
     \[
     11 - 9 = 2
     \]

5. **People who played only blackjack (B):**
   - 99 people played blackjack in total. We subtract the number of people who played blackjack and roulette (42), blackjack and craps (13), and all three games (9). So, the number of people who played only blackjack is:
     \[
     99 - 42 - 13 - 9 = 35
     \]

6. **People who played only roulette (R):**
   - 77 people played roulette in total. We subtract the number of people who played blackjack and roulette (42), roulette and craps (2), and all three games (9). So, the number of people who played only roulette is:
     \[
     77 - 42 - 2 - 9 = 24
     \]

7. **People who played only craps (C):**
   - 39 people played craps in total. We subtract the number of people who played blackjack and craps (13), roulette and craps (2), and all three games (9). So, the number of people who played only craps is:
     \[
     39 - 13 - 2 - 9 = 15
     \]

### Now, let's summarize the data for the Venn diagram:

- Only blackjack: **35**
- Only roulette: **24**
- Only craps: **15**
- Blackjack and roulette (but not craps): **42**
- Blackjack and craps (but not roulette): **13**
- Roulette and craps (but not blackjack): **2**
- All three games: **9**

### Venn Diagram Overview:

1. Total number of people surveyed: **160**
2. The sum of all people who participated in various games is:
   \[
   35 + 24 + 15 + 42 + 13 + 2 + 9 = 140
   \]
   This means that **20 people did not play any of the three games**.

Would you like me to create a visual representation of this Venn diagram?

---

Here are 5 related questions:

1. How would you calculate the number of people who played at least one game?
2. How can you verify if the survey data matches the total number of people surveyed?
3. What percentage of people played only one game?
4. How would you calculate the number of people who played exactly two games?
5. How can the inclusion-exclusion principle be applied to this scenario?

**Tip:** When dealing with Venn diagrams and set intersections, always start by filling in the innermost intersection (where all sets overlap) and work outward.

A survey of 160 people at a casino was conducted to determine the casino games that they typically play. The following data was collected. Complete the Venn Diagram. (Label each of the circles.) 99 people played blackjack, 77 people played roulette, 39 people played craps, 51 people played blackjack and roulette, 22 people played blackjack and craps, 11 played roulette and craps, and 9 people played blackjack, roulette, and craps.

Analyze the results of a survey of 160 casino visitors using a Venn diagram. Discover how many people played blackjack, roulette, craps, and combinations of these games using set theory and the inclusion-exclusion principle. Includes step-by-step calculations of intersections and exclusive game play counts. Suitable for Grades 10-12.

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