16. Unified State Exam in Anchuria. (7-11 grades, 2 points). In Anchuria, the Unified State Exam is taking place. The probability of guessing the correct answer to each question on the exam is 0.25. In 2011, to receive a certificate, one needed to answer correctly[^0]

3 questions out of 20. In 2012, the School Administration of Anchuria decided that 3 questions were too few. Now, one needs to answer correctly 6 questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of obtaining an Anchurian certificate higher - in 2011 or in 2012?
Solution. If a graduate guesses the answers, the Unified State Exam (EGE) can be considered as a Bernoulli scheme with a success probability of $p=0.25$ and a failure probability of $q=0.75$. In 2011, to pass the exam, one needed to answer at least 3 questions correctly. It is more convenient to find the probability of the opposite event, that is, the probability of answering fewer than 3 questions correctly. This probability is equal to the sum of the probabilities that the graduate answered 0, 1, and 2 questions correctly, that is, $\quad \sum_{k=0}^{2} C_{20}^{k} p^{k} q^{20-k}$. Using a spreadsheet, we find that this probability is approximately 0.091; therefore, the probability of passing the exam is approximately 0.909.

Similarly, we find that in 2012, the probability of failing the exam is

$$
\sum_{k=0}^{5} C_{40}^{k} p^{k} q^{40-k} \approx 0.043
$$

and the probability of passing the exam is approximately $0.957$.

Answer: In 2012, the probability of obtaining a certificate is higher.

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