When assessing the quality of binary classification, a confusion matrix is often used. Let's explain what it is. Suppose we have $$$N$$$ objects belonging to one of two classes (0 or 1). For example, the objects are X-ray images, where class 1 means the patient is sick, and class 0 means healthy. Let there be some algorithm that predicts the class of an object. Knowing the actual and predicted classes, we can construct the following table to evaluate the quality of the prediction:

Here, the value TP (True Positive) is the number of objects for which the model predicted class 1, and they actually belong to class 1. The value TN (True Negative) is the number of objects for which the model predicted class 0, and they actually belong to class 0. The value FP (False Positive) is the number of objects for which the model predicted class 1, but they actually belong to class 0. The value FN (False Negative) is the number of objects for which the model predicted class 0, but they actually belong to class 1.

From the values in this table, we can compute two more characteristics—precision and recall.

Precision is calculated using the formula $$$precision = TP / (TP + FP)$$$, which indicates the proportion of objects that our algorithm classified as class 1 that actually belong to class 1.

Recall is calculated using the formula $$$recall = TP / (TP + FN)$$$, which indicates the proportion of class 1 objects among all class 1 objects that our algorithm found.

Suppose there were a total of 99 objects, of which 35 are class 0 and 64 are class 1. It is known that the model correctly identified the class for 51 objects. Find the answers to the following four questions.

1. What is the maximum precision that could be achieved?
2. What is the minimum precision?
3. Suppose it is additionally known that precision equals recall. What is the maximum recall that could be achieved in this case?
4. What is the minimum recall?

In the answer, write four real numbers, each on a separate line. Use a point as the decimal separator. If you do not know any of the answers, write zeros instead. It is guaranteed that all numbers in the answer have a finite decimal part. In the "Language" field, select PHP (you do not need to know this language, it is just a feature of the checking system) and click the "Submit" button.

You are given a table with data on the evaluation of participants' solutions in a certain informatics olympiad: log.csv. A copy of this table is also available here: log.csv.

The table has 4 columns: problem — problem number (from 1 to 5); userId — participant identifier; language — programming language; score — points scored by the solution (from 0 to 100). The score values can also be empty (for example, if some solutions did not even compile on the server).

The olympiad was conducted according to traditional school rules: a participant can submit multiple solutions for each problem, and when summarizing the results, the best solution of the participant for each problem is selected, and the points for them are summed up.

Analyze the data in the table and answer the following 4 questions.

1. How many participants scored more than zero points?
2. How many participants submitted all their solutions in Python?
3. How many participants scored at least 200 points?
4. How many participants who scored at least 200 points submitted at least one solution in C++?

In your answer, write 4 integers, each on a separate line. Do not write anything extra. If you do not know some answers, write zeros instead. In the "Language" field, select PHP (you do not need to know this language, it is just a feature of the checking system) and click the "Submit" button.

A well-known medical clinic has decided to add treatment for a dangerous disease called "Inflammation of Cunning" to its list of services. However, before treating the disease, it must be diagnosed. To this end, the clinic commissioned three medical laboratories to develop test systems for performing the diagnosis. After some time, the laboratories provided their results.

It is known that all test systems use different technologies and algorithms to identify the disease. Therefore, in this problem, we will consider their diagnoses to be independent of each other (note: in practice, some dependence would still exist, but we will ignore it here).

The first test system makes a correct diagnosis with a probability of $$$p_1$$$, the second with a probability of $$$p_2$$$, and the third with a probability of $$$p_3$$$. The final decision is made based on the majority of the results. That is, if any two or all three test systems determine that the patient is sick, then the patient is considered sick; otherwise, they are not. Determine the probability that the clinic will make correct diagnoses.

Measures of central tendency are statistical indicators that describe the "typical" or central value in a dataset. There are several such measures; we will consider three of them.

The median of a set of numbers of odd length is the element that will be in the middle if the numbers are sorted. The mode is the most frequently occurring element. There can be multiple modes. The arithmetic mean is the sum of all elements divided by their count.

Write a program that finds a set of five integers that simultaneously satisfies the following three conditions:

1. the mode is equal to $$$x$$$, and it is unique;
2. the median is equal to $$$y$$$;
3. the arithmetic mean is equal to $$$z$$$.

Given $$$n$$$ points on a plane. Each point has coordinates $$$x_i$$$, $$$y_i$$$ and a color $$$c_i$$$, which can be white ($$$c_i=0$$$) or black ($$$c_i=1$$$).

To determine the color of a new point $$$P$$$, if it is unknown, the following approach can be used. The $$$k$$$ nearest points to it (where $$$k$$$ is some odd number) are taken, and the color that occurs most frequently among them is chosen. Note: the measure of proximity between two points can be calculated in different ways; in this problem, the usual Euclidean distance is used (i.e., the length of the segment between the points).

The quality of such a classifier depends on the choice of the parameter $$$k$$$. To find a good value for $$$k$$$, the following simple method can be used. We will look at what answers we get for our input points (for which the correct answers are known) with different $$$k$$$, and we will choose the value of $$$k$$$ for which the number of correct answers is maximized. This method is called "LOO cross-validation."

Let's explain how this method works in more detail. We will iterate through all odd values of $$$k$$$ in the range from $$$1$$$ to $$$n-1$$$. For each such $$$k$$$, we will iterate through all points. For each of them, we will find the $$$k$$$ nearest other points (not counting itself), take the color that occurs most frequently among them, and compare it with the actual color of that point. The more correct answers we get for the current value of $$$k$$$, the better that value is. Of course, to improve efficiency, the described implementation of the algorithm can be modified — the main thing is that the results remain correct.

Write a program that determines for each odd $$$k$$$ in the range from 1 to $$$n$$$ how many points will have their color correctly classified for the given $$$k$$$ during LOO cross-validation.

Vasya decided to develop a plugin for the new messenger Min. The plugin should determine whether an incoming message is fraudulent (i.e., written by scammers). Vasya has almost completed the work and implemented an algorithm that gives a score P for each message. The higher this score, the greater the likelihood that the message is fraudulent. The last detail remains — it is necessary to determine the triggering threshold, that is, starting from what threshold value T messages with a score P ≥ T will be considered fraudulent by the algorithm.

To solve this problem, Vasya formed a training dataset of N messages and enlisted the help of his classmates as experts. They read these N messages and marked which of them are fraudulent. All other messages are not fraudulent.

To evaluate the quality of the algorithm, Vasya decided to use the F₁ metric. Let's explain how it is calculated. We introduce the following notations:

• TP (True Positive) — the number of messages that the algorithm recognized as fraudulent and which are actually fraudulent.
• FP (False Positive) — the number of messages that the algorithm recognized as fraudulent, but which are actually not fraudulent.
• TN (True Negative) — the number of messages that the algorithm recognized as not fraudulent and which are actually not fraudulent.
• FN (False Negative) — the number of messages that the algorithm recognized as not fraudulent, but which are actually fraudulent.

Then the values of precision and recall are calculated using the formulas:

precision = TP / (TP + FP), which indicates what proportion of messages that our algorithm labeled as fraudulent are indeed fraudulent.

recall = TP / (TP + FN), which indicates what proportion of fraudulent messages among all fraudulent messages our algorithm found.

The F₁ metric is calculated using the formula:

Note: In the case when both precision and recall are zero, the resulting F₁ metric is also zero.

The larger the value of the F₁ metric, the higher the quality of classification. Find such a natural number — the threshold value T, which gives the maximum value of F₁.