(iii)A survey of 500 TV viewers produced the following information: 280 watch football, 150 watch hockey, 200 watch basketball, 70 watch football and basketball, 90 watch football and hockey, 80 watch hockey and basketball and 50 watch all the three games.Then,
(iii) How many viewers don't watch any game?

Answer

To find out how many viewers don't watch any game, we will use the principle of inclusion-exclusion. Let's denote:

- \( F \): Number of viewers who watch football = 280
- \( H \): Number of viewers who watch hockey = 150
- \( B \): Number of viewers who watch basketball = 200
- \( F \cap H \): Number of viewers who watch both football and hockey = 90
- \( F \cap B \): Number of viewers who watch both football and basketball = 70
- \( H \cap B \): Number of viewers who watch both hockey and basketball = 80
- \( F \cap H \cap B \): Number of viewers who watch all three games = 50

The formula for the union of three sets is:

\[ N(F \cup H \cup B) = N(F) + N(H) + N(B) - N(F \cap H) - N(F \cap B) - N(H \cap B) + N(F \cap H \cap B) \]

Now, substituting the values into the formula:

\[ N(F \cup H \cup B) = 280 + 150 + 200 - 90 - 70 - 80 + 50 \]

Calculating step by step:

1. Add the number of viewers for each game:
\[ 280 + 150 + 200 = 630 \]

2. Subtract the viewers who watch two games:
\[ 630 - 90 - 70 - 80 = 390 \]

3. Add the viewers who watch all three games:
\[ 390 + 50 = 440 \]

Thus, the number of viewers who watch at least one game is \( N(F \cup H \cup B) = 440 \).

Now, to find the number of viewers who don't watch any game:

\[ \text{Viewers who don't watch any game} = \text{Total viewers} - N(F \cup H \cup B) \]
\[ = 500 - 440 = 60 \]

Therefore, the number of viewers who don't watch any game is **60**.