Question

Квадратный лист бумаги со стороной 30 см разбивают на 900 квадратиков со стороной 1 см и среди эти квадратиков случайным образом выбирают один. Какова вероятность, что расстояние от любой из сторон выбранного квадратика до границы листа составит не менее 3 см?

Answer 1

По краю одной стороны расположено 30 квадратиков.
По краю всех 4 сторон 30*4-4 = 4*29 = 116 квадратиков.
Возьмем слой на 1 квадратик вглубь.
Вдоль одной стороны 28 квадратиков, вдоль всех 4 сторон 4*27 = 108.
Возьмем слой на 2 квадратика вглубь. 
Вдоль одной стороны 26 квадратиков, вдоль всех 4 сторон 4*25 = 100.
Возьмем слой на 3 квадратика вглубь. 
Вдоль одной стороны 24 квадратика, вдоль всех 4 сторон 4*23 = 92.
Это все квадратики, у которых расстояние до стороны меньше 3 см.
Их всего 116 + 108 + 100 + 92 = 416 квадратиков.
Остальных 900 - 416 = 484 квадратика.
Вероятность равна 484/900 = 121/225

Answer 2

Problem Analysis

We are given a square sheet of paper with a side length of 30 cm. This sheet is divided into 900 smaller squares, each with a side length of 1 cm. We randomly select one of these smaller squares. We need to find the probability that the distance from any side of the selected square to the edge of the sheet is at least 3 cm.

Solution

To solve this problem, we need to consider the possible positions of the selected square within the larger square sheet. Let's analyze the different cases:

1. Corner Squares: There are 4 corner squares in the larger square sheet. The distance from any side of these squares to the edge of the sheet is always 3 cm or more. Therefore, if the selected square is a corner square, the distance condition is satisfied.

2. Edge Squares: There are 88 edge squares in the larger square sheet. The distance from any side of these squares to the edge of the sheet is also always 3 cm or more. Therefore, if the selected square is an edge square, the distance condition is satisfied.

3. Interior Squares: The remaining squares are interior squares. For these squares, the distance from any side to the edge of the sheet can be less than 3 cm. To calculate the probability for these squares, we need to determine the number of interior squares and the total number of squares.

Calculation

Let's calculate the number of interior squares and the total number of squares:

- Number of interior squares: To find the number of interior squares, we need to subtract the number of corner squares (4) and the number of edge squares (88) from the total number of squares (900). Therefore, the number of interior squares is 900 - 4 - 88 = 808.

- Total number of squares: The total number of squares is given as 900.

Now, we can calculate the probability:

- Probability = Number of interior squares / Total number of squares - Probability = 808 / 900

Answer

The probability that the distance from any side of the selected square to the edge of the sheet is at least 3 cm is 808/900.