There are 20 rows labeled 1 to 20. There are 15 seats labeled 1 to 15 in each row. What is the probability that a person chosen is sitting in an odd-numbered row or in an odd-numbered seat?

To solve this, we need to find the probability that a person is sitting either in an odd-numbered row or an odd-numbered seat.

### Step 1: Total number of seats
There are 20 rows, each with 15 seats, so the total number of seats is:

\[
\text{Total seats} = 20 \times 15 = 300
\]

### Step 2: Counting the favorable outcomes

#### 1. **Odd-numbered rows**:
- The odd-numbered rows are: 1, 3, 5, ..., 19.
- There are 10 odd-numbered rows (half of 20).
- Each row has 15 seats, so the total number of seats in odd-numbered rows is:

\[
10 \times 15 = 150
\]

#### 2. **Odd-numbered seats**:
- The odd-numbered seats in each row are: 1, 3, 5, ..., 15.
- There are 8 odd-numbered seats in each row (since \( 1, 3, 5, \dots, 15 \) is a set of 8 numbers).
- Across all 20 rows, the total number of odd-numbered seats is:

\[
20 \times 8 = 160
\]

#### 3. **Seats counted twice (odd-numbered rows and odd-numbered seats)**:
- For seats that are both in odd-numbered rows and odd-numbered seats, we need to count how many odd-numbered seats are in the odd-numbered rows.
- There are 8 odd-numbered seats in each odd-numbered row, and there are 10 odd-numbered rows. So, the number of seats that fall in both categories is:

\[
10 \times 8 = 80
\]

### Step 3: Applying the inclusion-exclusion principle
To find the number of seats that are either in odd-numbered rows or odd-numbered seats, we apply the inclusion-exclusion principle:

\[
\text{Favorable outcomes} = (\text{Odd rows}) + (\text{Odd seats}) - (\text{Both odd rows and odd seats})
\]

\[
\text{Favorable outcomes} = 150 + 160 - 80 = 230
\]

### Step 4: Probability calculation
The probability is the ratio of favorable outcomes to total outcomes:

\[
\text{Probability} = \frac{230}{300} = \frac{23}{30}
\]

Thus, the probability that a person chosen is

Find the probability that a person is sitting in an odd-numbered row or an odd-numbered seat in a 20x15 grid. This problem explores the use of the inclusion-exclusion principle in probability, suitable for students in Grades 9-10.

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