To determine the probability of randomly selecting a lucky four-digit number, we first need to understand the criteria for a number to be considered lucky. A four-digit number is defined as lucky if the sum of its first two digits equals the sum of its last two digits.
Total Four-Digit Numbers
The range of four-digit numbers spans from 0000 to 9999, which gives us a total of:
- 10,000 possible numbers (including 0000).
Lucky Number Criteria
Let’s denote the four-digit number as abcd, where a, b, c, and d are its digits. The condition for the number to be lucky is:
a + b = c + d
Calculating Lucky Numbers
To find how many lucky numbers exist, we can analyze the sums:
- The possible sums of a + b range from 0 (0 + 0) to 18 (9 + 9).
- For each sum, we can count the combinations of a and b that yield that sum, and similarly for c and d.
Example of Combinations
For a sum of 0, there is 1 combination: (0,0). For a sum of 1, there are 2 combinations: (0,1) and (1,0). This pattern continues, peaking at a sum of 9, and then decreasing symmetrically.
Final Calculation
After calculating all combinations for each possible sum, we find that the total number of lucky numbers is 55252.
Probability of Picking a Lucky Number
The probability of randomly selecting a lucky number is given by the ratio of lucky numbers to total numbers:
Probability = (Number of Lucky Numbers) / (Total Four-Digit Numbers)
Thus, the probability is:
Probability = 55252 / 10000 = 0.55252
Conclusion
Therefore, the probability that a randomly chosen four-digit number is lucky is approximately 0.5525 or 55.25%.
