Question

В партии из 40 изделий 11 – первого сорта, 5 – второго сорта, 3 – третьего сорта. Остальные бракованные. Найдите вероятность того, что среди взятых 6 изделий найдется 2 – первого сорта, два – третьего сорта, одна бракованная?

Answer 1

Ответ:

Вероятность того, что наугад выбранные два изделия будут первого сорта равна .

Вероятность того, что случайно выбранные изделия будут второго сорта равна, равна

Вероятность того, что случайно выбранные изделия будут третьего сорта равна

По теореме сложения, вероятность того, что два случайно взятые изделия будут одного сорта равна

Answer 2

Calculating the Probability of Selecting Specific Items from a Batch

To calculate the probability of selecting 2 first-grade items, 2 third-grade items, and 1 defective item from a batch of 40 items with 11 first-grade, 5 second-grade, 3 third-grade, and the rest being defective, we can use the concept of combinations and probability.

The formula for calculating the probability of an event is:

Probability = (Number of favorable outcomes) / (Total number of outcomes)

In this case, the favorable outcomes are the combinations of selecting 2 first-grade items, 2 third-grade items, and 1 defective item from the batch of 40 items.

Applying the Formula

The total number of ways to choose 6 items out of 40 is given by the combination formula:

C(n, r) = n! / (r! * (n-r)!)

Where: - n = total number of items - r = number of items to be chosen

We can calculate the probability step by step using this formula.

Calculating the Probability

1. Number of ways to choose 2 first-grade items from 11: - C(11, 2) = 11! / (2! * (11-2)!) = 55

2. Number of ways to choose 2 third-grade items from 3: - C(3, 2) = 3! / (2! * (3-2)!) = 3

3. Number of ways to choose 1 defective item from the remaining (40-11-5-3) items: - C(21, 1) = 21

4. Total number of ways to choose 6 items from 40: - C(40, 6) = 40! / (6! * (40-6)!) = 3,838,380

5. Calculating the probability: - Probability = (55 * 3 * 21) / 3,838,380 ≈ 0.0007

Conclusion

The probability of selecting 2 first-grade items, 2 third-grade items, and 1 defective item from the batch of 40 items is approximately 0.0007.