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54 lines
2.6 KiB
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54 lines
2.6 KiB
HTML
<p>You have two soups, <strong>A</strong> and <strong>B</strong>, each starting with <code>n</code> mL. On every turn, one of the following four serving operations is chosen <em>at random</em>, each with probability <code>0.25</code> <strong>independent</strong> of all previous turns:</p>
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<ul>
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<li>pour 100 mL from type A and 0 mL from type B</li>
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<li>pour 75 mL from type A and 25 mL from type B</li>
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<li>pour 50 mL from type A and 50 mL from type B</li>
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<li>pour 25 mL from type A and 75 mL from type B</li>
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</ul>
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<p><strong>Note:</strong></p>
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<ul>
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<li>There is no operation that pours 0 mL from A and 100 mL from B.</li>
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<li>The amounts from A and B are poured <em>simultaneously</em> during the turn.</li>
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<li>If an operation asks you to pour <strong>more than</strong> you have left of a soup, pour all that remains of that soup.</li>
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</ul>
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<p>The process stops immediately after any turn in which <em>one of the soups</em> is used up.</p>
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<p>Return the probability that A is used up <em>before</em> B, plus half the probability that both soups are used up in the<strong> same turn</strong>. Answers within <code>10<sup>-5</sup></code> of the actual answer will be accepted.</p>
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<p> </p>
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<p><strong class="example">Example 1:</strong></p>
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<pre>
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<strong>Input:</strong> n = 50
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<strong>Output:</strong> 0.62500
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<strong>Explanation:</strong>
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If we perform either of the first two serving operations, soup A will become empty first.
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If we perform the third operation, A and B will become empty at the same time.
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If we perform the fourth operation, B will become empty first.
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So the total probability of A becoming empty first plus half the probability that A and B become empty at the same time, is 0.25 * (1 + 1 + 0.5 + 0) = 0.625.
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</pre>
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<p><strong class="example">Example 2:</strong></p>
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<pre>
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<strong>Input:</strong> n = 100
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<strong>Output:</strong> 0.71875
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<strong>Explanation:</strong>
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If we perform the first serving operation, soup A will become empty first.
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If we perform the second serving operations, A will become empty on performing operation [1, 2, 3], and both A and B become empty on performing operation 4.
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If we perform the third operation, A will become empty on performing operation [1, 2], and both A and B become empty on performing operation 3.
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If we perform the fourth operation, A will become empty on performing operation 1, and both A and B become empty on performing operation 2.
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So the total probability of A becoming empty first plus half the probability that A and B become empty at the same time, is 0.71875.
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</pre>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>0 <= n <= 10<sup>9</sup></code></li>
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</ul>
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