Question

Три станка выпускают болты. Первый дает 25%, второй - 35%, третий - 40% всей продукции. Причем, первый из своей выработанной продукции дает брак 5%, второй - 4%, третий - 2%. Выбранный болт оказался бракованным. Какова вероятность, что он произведен первой машиной?

Answer 1

ВЕРОЯТНОСТЬ РАВНА 0,1

Answer 2

Probability of the Bolt being Produced by the First Machine

To determine the probability that the defective bolt was produced by the first machine, we need to consider the overall production percentages and the defect rates of each machine.

According to the information provided: - The first machine produces 25% of the total production. - The second machine produces 35% of the total production. - The third machine produces 40% of the total production. - The defect rate of the first machine is 5%. - The defect rate of the second machine is 4%. - The defect rate of the third machine is 2%.

To calculate the probability, we can use Bayes' theorem. Bayes' theorem states that the probability of an event A given event B can be calculated as:

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

In this case, event A is the bolt being produced by the first machine, and event B is the bolt being defective.

Let's calculate the probability step by step:

1. Calculate the probability of the bolt being defective: - The bolt can be defective if it is produced by any of the three machines. - So, the probability of the bolt being defective is the sum of the defect rates of each machine multiplied by their respective production percentages: - (0.25 * 0.05) + (0.35 * 0.04) + (0.40 * 0.02) = 0.0125 + 0.014 + 0.008 = 0.0345

2. Calculate the probability of the bolt being produced by the first machine and being defective: - The probability of the bolt being produced by the first machine and being defective is the product of the defect rate of the first machine and its production percentage: - 0.25 * 0.05 = 0.0125

3. Calculate the probability of the bolt being produced by the first machine given that it is defective: - Using Bayes' theorem: - P(A|B) = (P(B|A) * P(A)) / P(B) - P(A|B) = (0.0125) / (0.0345) - P(A|B) ≈ 0.3623

Therefore, the probability that the defective bolt was produced by the first machine is approximately 0.3623 or 36.23%.