Question

Срочнооооооо. Станки №1,№2 и №3 с вероятностями брака соответственно 0,02,0,03,0,05 выпускают детали, поступающие на общий склад. Производительность станка №1 в 2 раза выше, чем №2 а станка № 2 в 3 раза выше, чем №3. Деталь, взятая со склада, оказалась бракованной. Какова вероятность,что она изготовлена на 1 станке?

Answer 1

Формула Байеса
Р= 0.6*0.02/(0.6*0.02+0.3*0.03+0.1*0.05)=6/13

Answer 2

Problem Analysis

We are given three machines, labeled as Machine #1, Machine #2, and Machine #3, with respective probabilities of producing defective parts as 0.02, 0.03, and 0.05. The productivity of Machine #1 is twice that of Machine #2, and the productivity of Machine #2 is three times that of Machine #3. We need to find the probability that a defective part is produced by Machine #1, given that a defective part is selected from the inventory.

Solution

To solve this problem, we can use Bayes' theorem. Bayes' theorem allows us to update our prior probability based on new evidence. In this case, the prior probability is the probability that a defective part is produced by Machine #1, and the evidence is the fact that a defective part is selected from the inventory.

Let's denote the events as follows: - A: The defective part is produced by Machine #1. - B: A defective part is selected from the inventory.

We need to find P(A|B), the probability that the defective part is produced by Machine #1 given that a defective part is selected from the inventory.

According to Bayes' theorem, we have:

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

Now, let's calculate the individual probabilities required for the formula.

1. P(B|A): This is the probability of selecting a defective part from the inventory given that it is produced by Machine #1. Since the defective rate for Machine #1 is given as 0.02, this probability is 0.02.

2. P(A): This is the prior probability of a defective part being produced by Machine #1. Since we are not given any information about the distribution of parts among the machines, we assume that each machine produces an equal number of parts. Therefore, the prior probability is 1/3.

3. P(B): This is the probability of selecting a defective part from the inventory. We can calculate this by considering the probabilities of defective parts produced by each machine and the productivity ratios between the