Question

У фокусника в шляпе 98 шаров. Фокусник при зрителях добавил в шляпу один красный и один синий шар. Затем он сказал, что будет доставать не глядя шары парами и в любой паре по крайней мере будет один синий шар. Сколько красных шаров в мешке?

Answer 1

В мешке 98 шаров
Добавили 1 красный и 1 синий
Что бы при доставании был как минимум 1 синий шар, в мешке должен
быть только один красный шар. Если будет 2 шара красных, то есть
вероятность их достать вместе, что противоречит заявлению. 
Всего в мешке 99 синих и 1 красный шар.

Answer 2

Problem Analysis

The magician starts with 98 balls in the hat. He adds one red ball and one blue ball to the hat. He then states that he will randomly select pairs of balls from the hat, and in each pair, there will be at least one blue ball. The question is how many red balls are in the hat.

Solution

To solve this problem, we need to consider the possible scenarios that satisfy the magician's statement. Let's analyze the possibilities step by step:

1. The magician selects a pair of balls from the hat. If there is at least one blue ball in the pair, the statement is satisfied. If there are no blue balls in the pair, the statement is not satisfied.

2. Let's assume that there are x red balls in the hat. Since the magician added one red ball and one blue ball, there are (x + 1) red balls and 2 blue balls in the hat.

3. We need to find the minimum value of x that satisfies the magician's statement. To do this, we need to consider the worst-case scenario, where the magician selects pairs of balls that do not contain any blue balls.

4. In the worst-case scenario, the magician selects pairs of balls in such a way that all the blue balls are selected before any red ball. This means that in each pair, the first ball selected is blue, and the second ball selected is red.

5. Let's assume that the magician selects y pairs of balls. In each pair, the first ball is blue, and the second ball is red. Since there are 2 blue balls in the hat, the maximum number of pairs the magician can select is 2.

6. In the worst-case scenario, the magician selects 2 pairs of balls, and in each pair, the second ball is red. This means that the magician selects 2 red balls from the hat.

7. Since the magician added one red ball to the hat, the total number of red balls in the hat is (2 + 1) = 3.

Answer

Therefore, there are 3 red balls in the hat.