A survey of 500 television watchers produced the following information, 285 watch foot-ball, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, 50 do not watch any of the three games. How many watch all the three games ? How many watch exactly one of the three games ?

Answer

To solve the problem step by step, we will use the principles of set theory and the information provided in the question.

### Step 1: Define the Sets
Let:
- \( F \) = set of people who watch football
- \( H \) = set of people who watch hockey
- \( B \) = set of people who watch basketball

From the problem, we have:
- \( |F| = 285 \) (number of football watchers)
- \( |H| = 195 \) (number of hockey watchers)
- \( |B| = 115 \) (number of basketball watchers)
- \( |F \cap B| = 45 \) (watch both football and basketball)
- \( |F \cap H| = 70 \) (watch both football and hockey)
- \( |H \cap B| = 50 \) (watch both hockey and basketball)
- 50 people do not watch any of the three games.

### Step 2: Calculate the Total Number of People Watching at Least One Game
The total number of people surveyed is 500. Since 50 do not watch any of the games, the number of people who watch at least one game is:
\[ |F \cup H \cup B| = 500 - 50 = 450 \]

### Step 3: Use the Principle of Inclusion-Exclusion
We will apply the formula for the union of three sets:
\[ |F \cup H \cup B| = |F| + |H| + |B| - |F \cap H| - |H \cap B| - |B \cap F| + |F \cap H \cap B| \]
Substituting the known values:
\[ 450 = 285 + 195 + 115 - 70 - 50 - 45 + |F \cap H \cap B| \]

### Step 4: Simplify the Equation
Calculating the right side:
\[ 450 = 285 + 195 + 115 - 70 - 50 - 45 + |F \cap H \cap B| \]
\[ 450 = 595 - 165 + |F \cap H \cap B| \]
\[ 450 = 430 + |F \cap H \cap B| \]

### Step 5: Solve for the Number of People Watching All Three Games
Now, isolate \( |F \cap H \cap B| \):
\[ |F \cap H \cap B| = 450 - 430 = 20 \]
Thus, **20 people watch all three games**.

### Step 6: Calculate the Number of People Watching Exactly One Game
To find the number of people who watch exactly one game, we can use the formula:
\[ \text{Exactly one} = |F| + |H| + |B| - 2(|F \cap H| + |H \cap B| + |B \cap F|) - 3|F \cap H \cap B| \]
Substituting the known values:
\[ \text{Exactly one} = 285 + 195 + 115 - 2(70 + 50 + 45) - 3(20) \]
Calculating:
\[ = 285 + 195 + 115 - 2(165) - 60 \]
\[ = 595 - 330 - 60 \]
\[ = 595 - 390 = 205 \]
Thus, **205 people watch exactly one of the three games**.

### Final Answers
- Number of people who watch all three games: **20**
- Number of people who watch exactly one of the three games: **205**