Two cards are drawn at random from 8 cards numbered from 1 to 8 . What is the probability that the sum of the numbers is odd , if the two cards are drawn together ?

Answer

To solve the problem of finding the probability that the sum of the numbers on two cards drawn from a set of 8 cards numbered from 1 to 8 is odd, we can follow these steps:

### Step 1: Identify the total number of outcomes
We need to find the total number of ways to choose 2 cards from 8 cards. This can be calculated using the combination formula:

\[ \text{Total Outcomes} = \binom{8}{2} \]

Calculating this gives:

\[ \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \]

### Step 2: Determine the favorable outcomes
The sum of two numbers is odd if one number is odd and the other is even.

- **Odd numbers from 1 to 8**: 1, 3, 5, 7 (Total = 4 odd numbers)
- **Even numbers from 1 to 8**: 2, 4, 6, 8 (Total = 4 even numbers)

To find the favorable outcomes, we need to choose one odd number and one even number.

The number of ways to choose one odd number from 4 odd numbers is:

\[ \text{Ways to choose 1 odd} = \binom{4}{1} = 4 \]

The number of ways to choose one even number from 4 even numbers is:

\[ \text{Ways to choose 1 even} = \binom{4}{1} = 4 \]

Thus, the total number of favorable outcomes (choosing one odd and one even) is:

\[ \text{Favorable Outcomes} = \binom{4}{1} \times \binom{4}{1} = 4 \times 4 = 16 \]

### Step 3: Calculate the probability
The probability \( P \) that the sum of the numbers is odd can be calculated using the formula:

\[ P(\text{sum is odd}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{16}{28} \]

Simplifying this fraction:

\[ P(\text{sum is odd}) = \frac{4}{7} \]

### Final Answer
The probability that the sum of the numbers on the two drawn cards is odd is:

\[ \frac{4}{7} \]