We are tasked with finding the probability of selecting a four-digit number made from the digits 0-7 that is less than or equal to 5000. The four-digit numbers cannot start with 0.

Step 1: Total possible four-digit numbers

The digits can be chosen from the set {0, 1, 2, 3, 4, 5, 6, 7}. The first digit cannot be 0, leaving 7 choices (1 through 7) for the first digit. For each of the remaining three digits, there are 8 possible choices (0 through 7).

Thus, the total number of possible four-digit numbers is: $7 \times 8 \times 8 \times 8 = 7 \times 8^3 = 7 \times 512 = 3584$

Step 2: Numbers less than or equal to 5000

For a number to be less than or equal to 5000, the first digit must be one of 1, 2, 3, 4, or 5 (since it cannot start with 0). We have 5 choices for the first digit. The remaining three digits can be any of the 8 digits.

Thus, the number of four-digit numbers less than or equal to 5000 is: $5 \times 8 \times 8 \times 8 = 5 \times 512 = 2560$

Step 3: Probability calculation

The probability is the ratio of the favorable outcomes (numbers less than or equal to 5000) to the total possible outcomes. Therefore, the probability is: $\text{Probability} = \frac{2560}{3584}$

We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 512: $\frac{2560}{3584} = \frac{5}{7}$

Thus, the probability is $\frac{5}{7}$.

Final Answer:

The probability of selecting a number less than or equal to 5000 is $\frac{5}{7}$, or approximately 0.7143 when rounded to four decimal places.

---

Would you like more details, or do you have any questions?

Here are some related questions you might find interesting:

1. What if the digits were chosen from the set {0-9} instead of {0-7}?
2. How would the probability change if the number had to be strictly less than 4000?
3. What if the number had to be divisible by 3?
4. How many of these numbers are prime?
5. How does the probability change if repetition of digits is not allowed?

Tip: When simplifying probabilities, always check if the numerator and denominator have a common factor!