Question

В партии из 15 деталей имеется 6 бракованных. Какова вероятность того, что среди наудачу отобранных 8 деталей окажутся 3 бракованные?

Answer 1

Всего 15 деталей, из них 9 доброкачественных и 6 бракованных.
Из 8 наудачу отобранных деталей должны быть 5 доброкачественных и
3 бракованных . Значит вероятность равна :
$\frac{5}{9}* \frac{3}{6} = \frac{5}{9}* \frac{1}{2}= \frac{5}{18}$

Answer 2

Calculating the Probability

To calculate the probability of selecting 3 defective items out of 8 randomly chosen from a batch of 15, where 6 items are defective, we can use the concept of combinations.

The probability can be calculated using the formula:

P = (Number of ways to choose 3 defective items out of 6) * (Number of ways to choose 5 non-defective items out of 9) / (Total number of ways to choose 8 items out of 15)

Let's calculate each part of the formula step by step.

1. Number of ways to choose 3 defective items out of 6: - This can be calculated using the combination formula: C(n, r) = n! / (r! * (n-r)!) - In this case, n = 6 (total number of defective items) and r = 3 (number of defective items to be chosen). - So, the number of ways to choose 3 defective items out of 6 is: C(6, 3) = 6! / (3! * (6-3)!) = 20

2. Number of ways to choose 5 non-defective items out of 9: - Since there are 15 items in total and 6 of them are defective, there are 9 non-defective items. - Using the combination formula, the number of ways to choose 5 non-defective items out of 9 is: C(9, 5) = 9! / (5! * (9-5)!) = 126

3. Total number of ways to choose 8 items out of 15: - This can also be calculated using the combination formula: C(15, 8) = 15! / (8! * (15-8)!) = 6435

Now, let's substitute these values into the formula to calculate the probability:

P = (Number of ways to choose 3 defective items out of 6) * (Number of ways to choose 5 non-defective items out of 9) / (Total number of ways to choose 8 items out of 15)

P = (20 * 126) / 6435 ≈ 0.392

Therefore, the probability that among the randomly selected 8 items, 3 of them will be defective is approximately 0.392 or 39.2%.