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54 lines
1.4 KiB
Python
54 lines
1.4 KiB
Python
"""
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Project Euler Problem 493: https://projecteuler.net/problem=493
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70 coloured balls are placed in an urn, 10 for each of the seven rainbow colours.
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What is the expected number of distinct colours in 20 randomly picked balls?
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Give your answer with nine digits after the decimal point (a.bcdefghij).
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-----
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This combinatorial problem can be solved by decomposing the problem into the
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following steps:
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1. Calculate the total number of possible picking cominations
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[combinations := binom_coeff(70, 20)]
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2. Calculate the number of combinations with one colour missing
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[missing := binom_coeff(60, 20)]
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3. Calculate the probability of one colour missing
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[missing_prob := missing / combinations]
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4. Calculate the probability of no colour missing
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[no_missing_prob := 1 - missing_prob]
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5. Calculate the expected number of distinct colours
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[expected = 7 * no_missing_prob]
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References:
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- https://en.wikipedia.org/wiki/Binomial_coefficient
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"""
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import math
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BALLS_PER_COLOUR = 10
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NUM_COLOURS = 7
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NUM_BALLS = BALLS_PER_COLOUR * NUM_COLOURS
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def solution(num_picks: int = 20) -> str:
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"""
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Calculates the expected number of distinct colours
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>>> solution(10)
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'5.669644129'
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>>> solution(30)
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'6.985042712'
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"""
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total = math.comb(NUM_BALLS, num_picks)
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missing_colour = math.comb(NUM_BALLS - BALLS_PER_COLOUR, num_picks)
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result = NUM_COLOURS * (1 - missing_colour / total)
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return f"{result:.9f}"
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if __name__ == "__main__":
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print(solution(20))
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