2023 AMC 10B Problem 21

Only the parities of the three bin counts matter, and each ball flips the parity of the bin it lands in. After $k$ balls the three parities sum to the parity of $k$, so after an odd number of balls the pattern is either "all odd" or "one odd, two even."

Call the pattern same when all three parities agree (all odd after an odd number of balls, all even after an even number) and split otherwise. Adding one ball moves a same pattern to split for sure, and moves a split pattern back to same with probability $\frac{1}{3}$ (the ball must land in the single bin whose parity differs).

Let $p_k$ be the probability of same after $k$ balls. Then
$$p_{k+1} = \tfrac{1}{3}(1 - p_k), \qquad p_1 = 0.$$
The fixed point is $p = \frac{1}{4}$, and $p_{k+1} - \frac{1}{4} = -\frac{1}{3}\left(p_k - \frac{1}{4}\right)$, so the error shrinks by a factor of $3$ every ball: $p_k = \frac{1}{4} - \frac{1}{4}\left(-\frac{1}{3}\right)^{k-1}$. For $k = 2023$ this is $\frac{1}{4}$ minus something astronomically small.

The answer is $\boxed{\textbf{(E)}\ \frac{1}{4}}$.

Reducing each bin to its parity turns $3^{2023}$ outcomes into a tiny state machine, and symmetry among the bins collapses it to two states. A random process that mixes (every state can reach every other) settles to its stationary distribution, here uniform over the four parity patterns allowed by an odd total, hence $\frac{1}{4}$. "Closest to" is the cue that a limit, not an exact value, is wanted.

Heuristic: for large $n$ the parities of bin $1$ and bin $2$ are each odd with probability about $\frac{1}{2}$ and are nearly independent, and bin $3$ is then odd automatically because the total is odd. So the probability is about $\frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4}$.

• Treating the three parities as independent coin flips or guessing 1/3, instead of noticing the odd total leaves four (nearly) equally likely parity patterns.
• Forgetting the constraint from the odd total, which makes "all even" impossible and leaves four, not eight, patterns to share the probability.