A box contains 100 balls numbered from 1 to 100. Three balls are selected at random with replacement. What is the probability that the sum of the three selected numbers is odd?

Introduction / Context:
This problem combines basic probability with parity reasoning. You draw three numbered balls from 1 to 100 with replacement and must determine the probability that the sum of the three numbers is odd. Instead of tracking each possible total, it is much easier to use the parity (odd or even) of each draw and apply simple rules for sums of odd and even numbers.

Given Data / Assumptions:

• The box contains the integers 1 through 100.
• There are 100 balls in total.
• Half of them, 50 numbers, are odd and the other 50 numbers are even.
• Three balls are selected with replacement, so each draw is independent.
• We want the probability that the sum of the three drawn numbers is odd.

Concept / Approach:
The parity of a sum depends on how many addends are odd. In particular:

• Even + even = even.
• Even + odd = odd.
• Odd + odd = even.

For three numbers, the sum is odd exactly when there are an odd number of odd terms. Therefore we need either exactly one odd and two evens, or exactly three odds. We first compute the probability of drawing an odd or even on a single draw, then use independence to find the probability of each pattern, and finally add these probabilities.

Step-by-Step Solution:
Step 1: Count odd and even numbers between 1 and 100. There are 50 odd numbers and 50 even numbers. Step 2: Probability of an odd number on a single draw = 50 / 100 = 1 / 2. Step 3: Probability of an even number on a single draw = 50 / 100 = 1 / 2. Step 4: Case 1: exactly one odd and two evens. Number of ways to choose which draw is odd = 3C1 = 3. Probability of a specific pattern like odd, even, even = (1/2) * (1/2) * (1/2) = 1/8. Total probability for exactly one odd = 3 * 1/8 = 3/8. Step 5: Case 2: all three numbers are odd. Probability of three odds in a row = (1/2) * (1/2) * (1/2) = 1/8. Step 6: The sum is odd if either Case 1 or Case 2 occurs, so add the probabilities. Total probability = 3/8 + 1/8 = 4/8 = 1/2.

Verification / Alternative check:
You can also argue by symmetry. Each draw is equally likely to be odd or even, and there is no bias toward producing odd or even sums when three independent symmetric draws are added. For every triple that sums to an odd number, you can create a corresponding triple that sums to an even number by flipping the parity of exactly one draw. This pairing shows that exactly half of all possible parity patterns produce an odd sum, giving a probability of 1/2, which matches the explicit calculation above.

Why Other Options Are Wrong:
Options 1/3 and 1/4 are too small and would imply that odd sums are much less common than even sums. Option 3/4 and option 2/3 are too large and would mean that odd sums dominate, which contradicts the symmetry argument. None of these values match the careful case analysis that shows 3/8 + 1/8 = 1/2.

Common Pitfalls:
A common error is to think that the sum is equally likely to be odd or even for any number of draws because each number is equally likely to be odd or even, without checking the parity rule. This is true for three draws but not for every possible count of draws, so relying on memory alone can be risky. Another mistake is to miscount the number of odd or even numbers between 1 and 100, but since they alternate, the count of 50 odd and 50 even is straightforward. Always link the parity of the sum to the number of odd terms, and then apply independence carefully.

Final Answer:
The probability that the sum of the three selected numbers is odd is 1/2.