시간 제한 | 메모리 제한 | 제출 | 정답 | 맞힌 사람 | 정답 비율 |
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2 초 | 1024 MB | 37 | 29 | 26 | 78.788% |
To qualify for cow camp, Bessie needs to earn a good score on the last problem of the USACOW Open contest. This problem has $T$ distinct test cases ($2\le T\le 10^3$) weighted equally, with the first test case being the sample case. Her final score will equal the number of test cases that her last submission passes.
Unfortunately, Bessie is way too tired to think about the problem, but since the answer to each test case is either "yes" or "no," she has a plan! Precisely, she decides to repeatedly submit the following nondeterministic solution:
if input == sample_input: print sample_output else: print "yes" or "no" each with probability 1/2, independently for each test case
Note that for all test cases besides the sample, this program may produce a different output when resubmitted, so the number of test cases that it passes will vary.
Bessie knows that she cannot submit more than $K$ ($1\le K\le 10^9$) times in total because then she will certainly be disqualified. What is the maximum possible expected value of Bessie's final score, assuming that she follows the optimal strategy?
2 3
1.875
In this example, Bessie should keep resubmitting until she has reached $3$ submissions or she receives full credit. Bessie will receive full credit with probability $\frac{7}{8}$ and half credit with probability $\frac{1}{8}$, so the expected value of Bessie's final score under this strategy is $\frac{7}{8}\cdot 2+\frac{1}{8}\cdot 1=\frac{15}{8}=1.875$. As we see from this formula, the expected value of Bessie's score can be calculated by taking the sum over $x$ of $p(x) \cdot x$, where $p(x)$ is the probability of receiving a score of $x$.
4 2
2.8750000000000000000
Here, Bessie should only submit twice if she passes fewer than $3$ test cases on her first try.