Question

Десять однотипных изделий, сходящих с конвейера, случайным способом распределяются по трем контейнерам, в каждое из которых может попасть любое количество этих изделий. Найдите вероятность того, что в первый контейнер попали 6 изделий, во второй — 3, а в третий — 1 изделие.

Answer 1

Ответ:

Здесь есть готовая формула: число перестановок с повторениями. То есть (6+3+1)!/(6!3!1!). Это число удачных расположений, и его надо разделить на число всех, то есть на 3^{10}.

Объяснение:

Answer 2

Probability of distributing 10 items randomly into 3 containers

To find the probability of distributing 10 items randomly into 3 containers, where each container can receive any number of items, we can use the concept of combinations.

The total number of ways to distribute the 10 items among the 3 containers is given by the formula:

Total number of ways = (n + r - 1) choose (r - 1)

Where: - n is the number of items (10 in this case) - r is the number of containers (3 in this case) - "choose" represents the binomial coefficient, which calculates the number of combinations

In this case, we have: - n = 10 - r = 3

Plugging these values into the formula, we get:

Total number of ways = (10 + 3 - 1) choose (3 - 1) = 12 choose 2 = 66

So, there are 66 different ways to distribute the 10 items among the 3 containers.

Probability of specific distribution

Now, let's find the probability of a specific distribution where 6 items go into the first container, 3 items go into the second container, and 1 item goes into the third container.

To calculate this probability, we need to determine the number of ways this specific distribution can occur and divide it by the total number of ways.

The number of ways this specific distribution can occur is given by the formula:

Number of ways = (6 choose 6) * (3 choose 3) * (1 choose 1)

Since choosing all the items from each container is only one way, the binomial coefficients become 1.

Therefore, the number of ways this specific distribution can occur is 1.

Now, we can calculate the probability:

Probability = Number of ways / Total number of ways = 1 / 66

So, the probability of having 6 items in the first container, 3 items in the second container, and 1 item in the third container is 1/66.

Please note that the calculations assume a random distribution of the items among the containers and do not take into account any specific constraints or preferences.