Lesson 4

Problem. Let us design a simple bot check.

The user has a score which starts at zero. The user receives test questions one by one. Each question has four possible answers, exactly one of them correct. For a correct answer the score increases by one, and for a wrong answer it decreases by one. After each question, if the score reaches \(3\), the user is declared human; if the score reaches \(-1\), the user is declared a bot.

How can we estimate the quality of such a system? Suppose a bot answers correctly with probability \(0.25\); in other words, it can only guess uniformly at random. For comparison, suppose a human answers correctly with probability \(0.95\).

Estimate four quantities:

1. the probability that the bot passes the check;
2. the average number of questions needed until the system makes a decision about this bot;
3. the probability that the human does not pass the check;
4. the average number of questions needed until the system makes a decision about this human.

Enter the four numbers separated by spaces, in the order listed above. Each probability will be accepted if its absolute error is at most \(0.01\); each expected number of questions will be accepted if its absolute error is at most \(0.1\).