Four digit numbers are formed by using the digits 1,2,3,4 and 5 without repeating any digit. Find the probability that a number, chosen at random, is an odd number.

Answer

To solve the problem of finding the probability that a randomly chosen four-digit number formed using the digits 1, 2, 3, 4, and 5 (without repetition) is an odd number, we can follow these steps:

### Step 1: Determine the total number of four-digit numbers
We can form a four-digit number using the digits 1, 2, 3, 4, and 5 without repetition.

- For the first digit, we can choose any of the 5 digits.
- For the second digit, we can choose from the remaining 4 digits.
- For the third digit, we can choose from the remaining 3 digits.
- For the fourth digit, we can choose from the remaining 2 digits.

Thus, the total number of ways to form a four-digit number is calculated as follows:

\[ \text{Total numbers} = 5 \times 4 \times 3 \times 2 = 120 \]

### Step 2: Determine the number of favorable outcomes (odd numbers)
A four-digit number is odd if its last digit is one of the odd digits available in our set, which are 1, 3, and 5.

1. **Choose the last digit**: We have 3 choices (1, 3, or 5).
2. **Choose the first three digits**: After selecting the last digit, we are left with 4 digits. We can choose any of these for the first three positions:
- For the first digit, we can choose from the remaining 4 digits.
- For the second digit, we can choose from the remaining 3 digits.
- For the third digit, we can choose from the remaining 2 digits.

Thus, the number of ways to form a four-digit odd number is calculated as follows:

\[ \text{Favorable outcomes} = 3 \times 4 \times 3 \times 2 = 72 \]

### Step 3: Calculate the probability
The probability of selecting an odd four-digit number is given by the ratio of the number of favorable outcomes to the total number of outcomes:

\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{72}{120} \]

### Step 4: Simplify the probability
To simplify \(\frac{72}{120}\):

\[ \frac{72 \div 24}{120 \div 24} = \frac{3}{5} \]

### Final Answer
The probability that a randomly chosen four-digit number formed using the digits 1, 2, 3, 4, and 5 is an odd number is:

\[ \frac{3}{5} \]