Question

В коробке находятся 45 ручек трех цветов, ручек каждого цвета поровну. Из коробки одну за другой вытягивают две ручки. Какова вероятность, что они одного цвета, если выбор осуществляется без возвращения?

Answer 1

15/45*14/44=1/3*7/22=7/66=0,106 (10,6%)

Answer 2

Problem Analysis

In this problem, we are given a box containing 45 pens of three different colors, with an equal number of pens for each color. We need to find the probability that when two pens are drawn from the box without replacement, they are of the same color.

Solution

To solve this problem, we can consider the different cases for drawing two pens of the same color. Since there are three colors, we will have three cases: drawing two pens of the first color, drawing two pens of the second color, and drawing two pens of the third color.

Let's calculate the probability for each case and then sum them up to find the overall probability.

Case 1: Drawing two pens of the first color

The probability of drawing the first pen of the first color is 15/45, as there are 15 pens of the first color out of a total of 45 pens. After drawing the first pen, there are 14 pens of the first color left out of a total of 44 pens. Therefore, the probability of drawing the second pen of the first color is 14/44.

The overall probability for this case is (15/45) * (14/44).

Case 2: Drawing two pens of the second color

The probability of drawing the first pen of the second color is also 15/45, as there are 15 pens of the second color out of a total of 45 pens. After drawing the first pen, there are 14 pens of the second color left out of a total of 44 pens. Therefore, the probability of drawing the second pen of the second color is 14/44.

The overall probability for this case is (15/45) * (14/44).

Case 3: Drawing two pens of the third color

The probability of drawing the first pen of the third color is 15/45, as there are 15 pens of the third color out of a total of 45 pens. After drawing the first pen, there are 14 pens of the third color left out of a total of 44 pens. Therefore, the probability of drawing the second pen of the third color is 14/44.

The overall probability for this case is (15/45) * (14/44).

Overall Probability

To find the overall probability, we need to sum up the probabilities from all three cases:

(15/45) * (14/44) + (15/45) * (14/44) + (15/45) * (14/44)

Simplifying this expression, we get:

(15 * 14 * 3) / (45 * 44)

Calculating this expression, we find that the overall probability is approximately 0.141.

Therefore, the probability that the two pens drawn are of the same color, without replacement, is approximately 0.141.

Conclusion

The probability that when two pens are drawn from the box without replacement, they are of the same color, is approximately 0.141.

Please note that the above solution assumes that the pens are well-mixed in the box and that the selection process is random.