Question

В ящике лежат 2 белых носка и4 серых носка и 7 черных носка. найдите вероятность того что из трех случайно выбранных носков хотя бы два окажутся одного цвета

Answer 1

Решение в картинке... Пишите, если где не можете понять или, если я ненароком ошибся)

Answer 2

всего 13 носков

пар:6 (2 носка одного цвета)
3/6=1/2=0,5 

Answer 3

Problem Analysis

In the given problem, we have a box containing 2 white socks, 4 gray socks, and 7 black socks. We need to find the probability that out of three randomly chosen socks, at least two socks will be of the same color.

Solution

To solve this problem, we can consider the following cases: 1. Case 1: Two socks of the same color and one sock of a different color. 2. Case 2: Three socks of the same color.

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

Case 1: Two socks of the same color and one sock of a different color

To calculate the probability for this case, we need to consider all possible combinations of colors for the two socks and the remaining sock.

1. Two white socks and one sock of a different color: - Number of ways to choose two white socks: C(2, 2) = 1 - Number of ways to choose one sock of a different color: C(11, 1) = 11 - Total number of ways to choose three socks: C(13, 3) = 286 - Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = (1 * 11) / 286

2. Two gray socks and one sock of a different color: - Number of ways to choose two gray socks: C(4, 2) = 6 - Number of ways to choose one sock of a different color: C(9, 1) = 9 - Total number of ways to choose three socks: C(13, 3) = 286 - Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = (6 * 9) / 286

3. Two black socks and one sock of a different color: - Number of ways to choose two black socks: C(7, 2) = 21 - Number of ways to choose one sock of a different color: C(9, 1) = 9 - Total number of ways to choose three socks: C(13, 3) = 286 - Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = (21 * 9) / 286

Now, let's calculate the probability for Case 1 by adding the probabilities of the three sub-cases.

Case 2: Three socks of the same color

To calculate the probability for this case, we need to consider all possible colors and calculate the probability for each color.

1. Three white socks: - Number of ways to choose three white socks: C(2, 3) = 0 - Probability = 0

2. Three gray socks: - Number of ways to choose three gray socks: C(4, 3) = 4 - Total number of ways to choose three socks: C(13, 3) = 286 - Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = 4 / 286

3. Three black socks: - Number of ways to choose three black socks: C(7, 3) = 35 - Total number of ways to choose three socks: C(13, 3) = 286 - Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = 35 / 286

Now, let's calculate the probability for Case 2 by adding the probabilities of the three sub-cases.

Overall Probability

To find the overall probability, we need to add the probabilities of Case 1 and Case 2.

Overall Probability = Probability of Case 1 + Probability of Case 2

Now, let's calculate the probabilities and find the overall probability.

Calculation

Case 1: - Probability for two white socks and one sock of a different color: (1 * 11) / 286 - Probability for two gray socks and one sock of a different color: (6 * 9) / 286 - Probability for two black socks and one sock of a different color: (21 * 9) / 286

Case 2: - Probability for three white socks: 0 - Probability for three gray socks: 4 / 286 - Probability for three black socks: 35 / 286

Overall Probability = Probability of Case 1 + Probability of Case 2

Now, let's calculate the overall probability.

Calculation

Overall Probability = ((1 * 11) / 286) + ((6 * 9) / 286) + ((21 * 9) / 286) + (0) + (4 / 286) + (35 / 286)

Overall Probability = (11 + 54 + 189 + 0 + 4 + 35) / 286

Overall Probability = 293 / 286

Overall Probability ≈ 1.024

Therefore, the probability that out of three randomly chosen socks, at least two socks will be of the same color is approximately 1.024.