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39 lines
1.4 KiB
Python
39 lines
1.4 KiB
Python
def expected_black_ball_position(N: int, K: int) -> int:
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""" There are N-1 white balls and one black ball arranged in a row, with the black ball
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initially at the leftmost position. Takahashi performs K operations, where each operation
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consists of:
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- Choose two integers a and b uniformly at random between 1 and N, inclusive
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- If a ≠ b, swap the a-th and b-th balls from the left
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Find the expected position of the black ball after K operations, modulo 998244353.
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The result is returned as an integer R where R × Q ≡ P (mod 998244353), where P/Q is
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the expected value expressed as an irreducible fraction.
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Args:
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N: Total number of balls (1 ≤ N ≤ 998244352)
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K: Number of operations (1 ≤ K ≤ 10^5)
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Returns:
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The expected position modulo 998244353
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>>> expected_black_ball_position(2, 1)
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499122178
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>>> expected_black_ball_position(3, 2)
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554580198
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>>> expected_black_ball_position(4, 4)
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592707587
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"""
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MOD = 998244353
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if N == 1:
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return 1
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Nm = N % MOD
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inv_N = pow(Nm, MOD - 2, MOD)
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# a = 1 - 2/N : factor by which the "excess" probability mass decays
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a = (1 - 2 * inv_N) % MOD
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aK = pow(a, K, MOD)
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# E = ((N+1) - (N-1) * a^K) / 2 (mod MOD)
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num = ((Nm + 1) - ((Nm - 1) * aK)) % MOD
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inv2 = pow(2, MOD - 2, MOD)
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return num * inv2 % MOD
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