Checkpoint experiments through SWE pilot v2

results/board-interface-v2-sept8/reviews/board-c2-task11-submitted.py

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+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
+    if N == 1:
+        return 1
+    Nm = N % MOD
+    inv_N = pow(Nm, MOD - 2, MOD)
+    # a = 1 - 2/N : factor by which the "excess" probability mass decays
+    a = (1 - 2 * inv_N) % MOD
+    aK = pow(a, K, MOD)
+    # E = ((N+1) - (N-1) * a^K) / 2  (mod MOD)
+    num = ((Nm + 1) - ((Nm - 1) * aK)) % MOD
+    inv2 = pow(2, MOD - 2, MOD)
+    return num * inv2 % MOD