Checkpoint experiments through SWE pilot v2

results/board-muse-sept8/artifact-replays/episode-008/func.py

@@ -0,0 +1,38 @@
+def expected_black_ball_position(N: int, K: int) -> int:
+    """ There are N-1 white balls and one black ball arranged in a row, with the black ball
+    initially at the leftmost position. Takahashi performs K operations, where each operation
+    consists of:
+    - Choose two integers a and b uniformly at random between 1 and N, inclusive
+    - If a ≠ b, swap the a-th and b-th balls from the left
+    
+    Find the expected position of the black ball after K operations, modulo 998244353.
+    
+    The result is returned as an integer R where R × Q ≡ P (mod 998244353), where P/Q is
+    the expected value expressed as an irreducible fraction.
+    
+    Args:
+        N: Total number of balls (1 ≤ N ≤ 998244352)
+        K: Number of operations (1 ≤ K ≤ 10^5)
+    
+    Returns:
+        The expected position modulo 998244353
+    
+    >>> expected_black_ball_position(2, 1)
+    499122178
+    >>> expected_black_ball_position(3, 2)
+    554580198
+    >>> expected_black_ball_position(4, 4)
+    592707587
+    """
+    MOD = 998244353
+    inv2 = (MOD + 1) // 2  # 499122177
+    # Expected value: ((N+1) - (N-1)*((N-2)/N)^K) / 2
+    # Derived from symmetry: q_K = 1/N + ((N-2)/N)^K * (N-1)/N
+    # E_K = (N+1)/2 - (N-1)/2 * ((N-2)/N)^K ... actually
+    # E_K = (N+1)/2 - (N-1)/2 * r^K with r=(N-2)/N
+    # Equivalent to ((N+1) - (N-1)*r^K)/2
+    n_mod = N % MOD
+    inv_n = pow(n_mod, MOD - 2, MOD)
+    r = (n_mod - 2) % MOD * inv_n % MOD
+    p = pow(r, K, MOD)
+    return ((n_mod + 1 - (n_mod - 1) * p) % MOD) * inv2 % MOD