Question

Независимо друг от друга3 человека садятся в по­езд, содержащий 12 вагонов. Найдите вероятность того, что по крайней мере двое из них окажутся в одном вагоне.

Answer 1

Probability of at least two people being in the same train car

To find the probability that at least two out of three people end up in the same train car, we can use the concept of complementary probability. We will calculate the probability that all three people are in different train cars and then subtract it from 1 to get the desired probability.

Let's break down the problem step by step:

1. Total number of ways to seat three people in twelve train cars: - Since each person can be seated in any of the twelve train cars, the total number of ways to seat them is 12 * 12 * 12 = 1,728.

2. Number of ways to seat three people in different train cars: - For the first person, there are twelve options to choose from. - For the second person, there are eleven remaining options. - For the third person, there are ten remaining options. - Therefore, the number of ways to seat them in different train cars is 12 * 11 * 10 = 1,320.

3. Probability of three people being in different train cars: - The probability is calculated by dividing the number of favorable outcomes (seating three people in different train cars) by the total number of possible outcomes (seating three people in any train car). - So, the probability is 1,320 / 1,728 ≈ 0.764.

4. Probability of at least two people being in the same train car: - Since this is the complementary probability, we subtract the probability of three people being in different train cars from 1. - Therefore, the probability is 1 - 0.764 = 0.236, or approximately 23.6%.

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