Four digit numbers are formed by using the digits 1,2,3,4 and 5 without repeating the 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 repeating any digit) is an odd number, we can follow these steps:

### Step 1: Calculate the Total Number of Four-Digit Numbers
We can form a four-digit number using the digits 1, 2, 3, 4, and 5. The first digit can be any of the 5 digits. Once the first digit is chosen, we have 4 remaining digits for the second digit, 3 remaining for the third digit, and 2 remaining for the fourth digit.

So, the total number of four-digit numbers can be calculated as:
\[ \text{Total Numbers} = 5 \times 4 \times 3 \times 2 = 120 \]

### Step 2: Identify the Criteria for Odd Numbers
For a number to be odd, its last digit must be one of the odd digits available in our set, which are 1, 3, and 5.

### Step 3: Calculate the Number of Favorable Outcomes (Odd Numbers)
1. **Choose the last digit**: We have 3 choices (1, 3, or 5).
2. **Choose the first digit**: After choosing the last digit, we have 4 remaining digits to choose from for the first digit.
3. **Choose the second digit**: After choosing the first digit, we have 3 remaining digits.
4. **Choose the third digit**: After choosing the second digit, we have 2 remaining digits.

Thus, the number of ways to form an odd four-digit number is:
\[ \text{Favorable Outcomes} = 3 \times 4 \times 3 \times 2 = 72 \]

### Step 4: Calculate the Probability
The probability of choosing an odd 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 5: 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 is an odd number is:
\[ \frac{3}{5} \]

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