A bag contains tickets numbered 1 to 20 . Two tickets are drawn at random , Find the probability that sum of the two numbers on the tickets is even .

Answer

To find the probability that the sum of the two numbers on the tickets drawn from a bag containing tickets numbered 1 to 20 is even, we can follow these steps:

### Step 1: Identify the Total Number of Tickets
The tickets are numbered from 1 to 20. Therefore, the total number of tickets is:
\[ N = 20 \]

### Step 2: Determine the Total Ways to Draw Two Tickets
The total number of ways to choose 2 tickets from 20 is given by the combination formula:
\[ \text{Total ways} = \binom{20}{2} = \frac{20 \times 19}{2 \times 1} = 190 \]

### Step 3: Identify Even and Odd Tickets
From 1 to 20:
- Even numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 (10 even numbers)
- Odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 (10 odd numbers)

### Step 4: Calculate Ways to Choose Two Even Numbers
The number of ways to choose 2 even numbers from the 10 even numbers is:
\[ \text{Ways to choose 2 even} = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]

### Step 5: Calculate Ways to Choose Two Odd Numbers
Similarly, the number of ways to choose 2 odd numbers from the 10 odd numbers is:
\[ \text{Ways to choose 2 odd} = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]

### Step 6: Calculate Total Favorable Outcomes
The total number of favorable outcomes (either both numbers are even or both numbers are odd) is:
\[ \text{Total favorable outcomes} = \text{Ways to choose 2 even} + \text{Ways to choose 2 odd} = 45 + 45 = 90 \]

### Step 7: Calculate the Probability
The probability \( P \) that the sum of the two numbers is even is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Total favorable outcomes}}{\text{Total ways}} = \frac{90}{190} = \frac{9}{19} \]

### Final Answer
The probability that the sum of the two numbers on the tickets is even is:
\[ \boxed{\frac{9}{19}} \]