Choosing two different number from first three natural numbers, the probability of one of the numbers to be maximum of three is :

Options

A. 1
B. `1//3`
C. `2//3`
D. `1//2`

Answer

To solve the problem of finding the probability that one of the chosen numbers is the maximum of three when selecting two different numbers from the first three natural numbers, we can follow these steps:

### Step 1: Identify the first three natural numbers
The first three natural numbers are 1, 2, and 3.

### Step 2: Determine the possible pairs
When choosing two different numbers from the set {1, 2, 3}, the possible pairs are:
- (1, 2)
- (1, 3)
- (2, 3)

### Step 3: Identify the maximum in each pair
Now, we will identify the maximum number in each of these pairs:
- In the pair (1, 2), the maximum is 2.
- In the pair (1, 3), the maximum is 3.
- In the pair (2, 3), the maximum is 3.

### Step 4: Count the favorable outcomes
Next, we count how many of these pairs have the maximum number being 3:
- The pairs (1, 3) and (2, 3) both have 3 as the maximum.

Thus, there are 2 favorable outcomes where one of the chosen numbers is the maximum (3).

### Step 5: Calculate the total outcomes
The total number of ways to choose 2 different numbers from 3 is given by the combination formula \( C(n, r) \), where \( n \) is the total numbers to choose from, and \( r \) is the number of choices made:
\[ C(3, 2) = 3 \]
So, there are 3 total outcomes.

### Step 6: Calculate the probability
The probability \( P \) that one of the chosen numbers is the maximum (which is 3) can be calculated using the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]
Substituting the values we found:
\[ P = \frac{2}{3} \]

### Final Answer
The probability that one of the chosen numbers is the maximum of three is \( \frac{2}{3} \).

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