If without repetition of the numbers, four digit numbers are formed with the number 0, 2,3, and 5, then find the probability of such number being divisible by 5.

Answer

To solve the problem of finding the probability that a four-digit number formed from the digits 0, 2, 3, and 5 is divisible by 5, we can follow these steps:

### Step 1: Determine the Total Number of Four-Digit Numbers
To form a four-digit number using the digits 0, 2, 3, and 5 without repetition, we need to ensure that the first digit is not 0 (as it would then be a three-digit number).

1. **Choose the first digit**: It can be either 2, 3, or 5 (3 options).
2. **Choose the second digit**: After choosing the first digit, we have 3 digits left (including 0), so we have 3 options.
3. **Choose the third digit**: After choosing the first and second digits, we have 2 digits left, so we have 2 options.
4. **Choose the fourth digit**: After choosing the first three digits, we have 1 digit left, so we have 1 option.

Thus, the total number of four-digit numbers is:
\[ \text{Total Numbers} = 3 \times 3 \times 2 \times 1 = 18 \]

### Step 2: Determine the Number of Favorable Outcomes (Divisible by 5)
A number is divisible by 5 if it ends in either 0 or 5. We will consider both cases separately.

#### Case 1: Last Digit is 0
- The first digit can be either 2, 3, or 5 (3 options).
- The second digit can be filled with the remaining 2 digits (2 options).
- The third digit can be filled with the last remaining digit (1 option).

Thus, the total for this case is:
\[ \text{Case 1 Total} = 3 \times 2 \times 1 = 6 \]

#### Case 2: Last Digit is 5
- The first digit can only be 2 or 3 (2 options), since it cannot be 0.
- The second digit can be filled with the remaining 2 digits (including 0) (2 options).
- The third digit can be filled with the last remaining digit (1 option).

Thus, the total for this case is:
\[ \text{Case 2 Total} = 2 \times 2 \times 1 = 4 \]

### Step 3: Calculate the Total Favorable Outcomes
Now, we add the totals from both cases:
\[ \text{Total Favorable Outcomes} = 6 + 4 = 10 \]

### Step 4: Calculate the Probability
The probability \( P \) that a randomly formed four-digit number is divisible by 5 is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P = \frac{\text{Total Favorable Outcomes}}{\text{Total Numbers}} = \frac{10}{18} = \frac{5}{9} \]

### Final Answer
Thus, the probability that a four-digit number formed from the digits 0, 2, 3, and 5 is divisible by 5 is:
\[ \frac{5}{9} \]