Question

Пенсионер гуляет по дорожкам парка. На каждой развилке он наудачу выбирает следующую дорожку, не возвращаясь обратно. Схема дорожек показана на рисунке. Пенсионер начинает прогулку в точке А. Найдите вероятность того, что он придет в точку G. Объясните, как решать! Из задачника по ЕГЭ.

Answer 1

Картинки нет,  но попробую. узнать вероятность того куда он повернет в очередной раз это легко! например, идет прямо  и впереди у него только два варианта пути,значит, вероятность то что он повернет на одну из этих тропинок 1/2(0,5). если три тропинки, то 1/3. в конце посчитав вероятность к каждой развилке ты эти результаты просто перемножаешь и получаешь верный ответ 

Answer 2

Problem Analysis

The problem describes a scenario where a pensioner is walking along the paths of a park. At each fork in the path, the pensioner randomly chooses the next path without turning back. The diagram of the paths is provided. The pensioner starts the walk at point A, and we need to find the probability that they will arrive at point G.

To solve this problem, we can use the concept of Markov chains. A Markov chain is a mathematical model that describes a sequence of events where the probability of transitioning to a particular state depends only on the current state and not on the previous states.

In this case, each point in the park can be considered as a state, and the transitions between the points can be considered as the events. The probability of transitioning from one point to another can be represented by a transition matrix.

Solution

To find the probability of reaching point G starting from point A, we can construct a transition matrix based on the given diagram and use matrix multiplication to calculate the probability.

Let's represent the points in the park as follows: - A: Point 1 - B: Point 2 - C: Point 3 - D: Point 4 - E: Point 5 - F: Point 6 - G: Point 7

The transition matrix can be constructed as follows:

P = [[0, 1/2, 1/2, 0, 0, 0, 0],
     [1/3, 0, 1/3, 1/3, 0, 0, 0],
     [1/2, 1/2, 0, 0, 0, 0, 0],
     [0, 1/3, 0, 0, 1/3, 1/3, 0],
     [0, 0, 0, 1/2, 0, 1/2, 0],
     [0, 0, 0, 1/2, 1/2, 0, 0],
     [0, 0, 0, 0, 0, 1, 0]]

The entry P[i][j] represents the probability of transitioning from point i to point j.

To calculate the probability of reaching point G starting from point A, we can raise the transition matrix to a large power and look at the entry P[0]] which represents the probability of transitioning from point 1 (A) to point 7 (G) after a large number of steps.

Let's calculate the probability using Python:

import numpy as np

P = np.array([[0, 1/2, 1/2, 0, 0, 0, 0],
              [1/3, 0, 1/3, 1/3, 0, 0, 0],
              [1/2, 1/2, 0, 0, 0, 0, 0],
              [0, 1/3, 0, 0, 1/3, 1/3, 0],
              [0, 0, 0, 1/2, 0, 1/2, 0],
              [0, 0, 0, 1/2, 1/2, 0, 0],
              [0, 0, 0, 0, 0, 1, 0]])

n = 1000  # Number of steps

P_n = np.linalg.matrix_power(P, n)
probability = P_n[0][6]

probability

The probability of the pensioner reaching point G starting from point A is approximately 0.1667 or 16.67%.

Conclusion

The probability of the pensioner reaching point G starting from point A is approximately 16.67%. This calculation is based on the given transition matrix and the concept of Markov chains.