Question

54. Два працівники виготовили однакову кількість деталей. Ймовірність того, що перший зробить браковану деталь рівна0,05; другий - 0,1. При перевірці була виявлена бракована деталь. Знайти ймовірність того, що деталь була виготовлена а) першим б)другим працівником

Answer 1

Ответ:

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Answer 2

Problem Analysis

We have two workers who have produced an equal number of parts. The probability that the first worker produces a defective part is 0.05, while the probability that the second worker produces a defective part is 0.1. During inspection, a defective part is found. We need to find the probability that the defective part was produced by the first worker and the second worker.

Solution

To solve this problem, we can use Bayes' theorem. Bayes' theorem allows us to update our prior probabilities based on new evidence. In this case, the evidence is the defective part found during inspection.

Let's define the following events: - A: The part is defective. - B1: The part was produced by the first worker. - B2: The part was produced by the second worker.

We need to find the probability of event B1 given event A (i.e., the probability that the part was produced by the first worker given that it is defective). Similarly, we need to find the probability of event B2 given event A (i.e., the probability that the part was produced by the second worker given that it is defective).

According to Bayes' theorem, the probability of event B1 given event A can be calculated as follows:

P(B1|A) = (P(A|B1) * P(B1)) / P(A)

Similarly, the probability of event B2 given event A can be calculated as follows:

P(B2|A) = (P(A|B2) * P(B2)) / P(A)

To calculate these probabilities, we need to know the prior probabilities P(B1) and P(B2), as well as the conditional probabilities P(A|B1) and P(A|B2).

Given that the first worker has a probability of 0.05 of producing a defective part and the second worker has a probability of 0.1, we can assign the following prior probabilities:

- P(B1) = 0.5 (since both workers have produced an equal number of parts) - P(B2) = 0.5 (since both workers have produced an equal number of parts)

We also know the conditional probabilities:

- P(A|B1) = 0.05 (the probability of finding a defective part given that it was produced by the first worker) - P(A|B2) = 0.1 (the probability of finding a defective part given that it was produced by the second worker)

Now, let's calculate the probabilities using the given values:

P(B1|A) = (P(A|B1) * P(B1)) / P(A) = (0.05 * 0.5) / P(A)

P(B2|A) = (P(A|B2) * P(B2)) / P(A) = (0.1 * 0.5) / P(A)

To find the value of P(A), we need to consider all possible ways in which event A can occur. In this case, event A can occur if the part is defective and it was produced by either the first worker or the second worker. Therefore, we can calculate P(A) as follows:

P(A) = P(A|B1) * P(B1) + P(A|B2) * P(B2) = (0.05 * 0.5) + (0.1 * 0.5)

Now, let's substitute the values into the equations to find the probabilities:

P(B1|A) = (0.05 * 0.5) / [(0.05 * 0.5) + (0.1 * 0.5)]

P(B2|A) = (0.1 * 0.5) / [(0.05 * 0.5) + (0.1 * 0.5)]

Simplifying the equations:

P(B1|A) = 0.025 / 0.075 = 1/3

**P(B2|A) = 0.05 / 0