A box contains 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:
This probability problem focuses on the parity (odd or even) of the sum of numbers chosen from a set. Instead of analysing all specific values from 1 to 100, we can exploit symmetry between odd and even numbers and use rules about when sums are odd or even. This greatly simplifies the calculation.

Given Data / Assumptions:

• Balls are numbered 1 to 100.
• Each draw is random and with replacement (so the distribution remains the same for each draw).
• We draw three balls: say numbers X₁, X₂, X₃.
• We need the probability that X₁ + X₂ + X₃ is odd.

Concept / Approach:
Parity rules:
Even + Even = Even, Odd + Odd = Even, Odd + Even = Odd.For the sum of three numbers to be odd, the total number of odd addends must itself be odd (i.e., 1 or 3). Also, from 1 to 100, there are 50 odd numbers and 50 even numbers, so each draw has probability 1/2 of being odd and 1/2 of being even. We treat each draw as an independent Bernoulli trial with "odd" as success with probability 1/2.

Step-by-Step Solution:
Let p = probability that a single draw is odd.From 1 to 100, half the numbers are odd, so p = 50/100 = 1/2.Probability of even = 1 − p = 1/2.We want sum to be odd. This happens when the number of odd draws among the three is 1 or 3.Case 1: Exactly one odd and two even numbers.Probability = C(3, 1) * (1/2)¹ * (1/2)² = 3 * (1/8) = 3/8.Case 2: All three numbers are odd.Probability = (1/2)³ = 1/8.Total probability of an odd sum = 3/8 + 1/8 = 4/8 = 1/2.

Verification / Alternative check:
Another neat observation is that if each draw has equal chance of being odd or even, then the distribution of the sum’s parity is symmetric: exactly half of all possible parity combinations (odd-odd-odd, odd-even-even, etc.) yield an odd sum. This symmetry argument also leads to the result 1/2 without detailed counting.

Why Other Options Are Wrong:
1/6, 1/3 and 1/4 are too small and would correspond to far fewer favourable combinations than actually exist. 2/3 is too large, implying that odd sums are significantly more likely than even sums, which contradicts the symmetry in odd and even probabilities. Only 1/2 correctly reflects the equal likelihood of odd and even sums under these conditions.

Common Pitfalls:
Common mistakes include trying to work with actual numbers from 1 to 100 instead of just using odd/even counts, or forgetting that the draws are with replacement, which keeps probabilities constant for each draw. Another error is to consider only the all-odd case and forget the one-odd-two-even case, or vice versa.

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