Question

In a certain factory, machines I, II, and III are all producing springs of the same length. Machines I, II, and III produce \(1 \%, 4 \%\), and \(2 \%\) defective springs, respectively. Of the total production of springs in the factory, Machine I produces \(30 \%\), Machine II produces \(25 \%\), and Machine III produces \(45 \%\). (a) If one spring is selected at random from the total springs produced in a given day, determine the probability that it is defective. (b) Given that the selected spring is defective, find the conditional probability that it was produced by Machine II.

Short answer

The probability that a randomly selected spring is defective is 2.2%. Given that a spring is defective, the probability that it was produced by Machine II is approximately 45.45%.

Step-by-step solution

Find the total probability of a defective spring

We are given that Machine I produces 30% of the total springs and 1% of these are defective. So its contribution to the total defective springs is 0.30 * 0.01 = 0.003 (or 0.3%). Similarly, Machine II and III contribute 0.25 * 0.04 = 0.01 (or 1%) and 0.45 * 0.02 = 0.009 (or 0.9%) respectively. Adding these up gives the total probability of a defective spring.

Calculation

Adding the contributions, we get: 0.003 + 0.01 + 0.009 = 0.022 or 2.2%.

Find the conditional probability that the defective spring was made by Machine II

We first find the probability that a spring is defective given that it was made by Machine II which is 4%. This is denoted as P(D|M2). We also know the probability that a spring was made by Machine II irrespective of it being defective or not, which is 25%. This can be written as P(M2). We found the total probability of a defective spring (from steps 1 and 2) which is denoted as P(D). The conditional probability that the spring was made by Machine II given it is defective, can be found using Bayes' theorem: P(M2|D) = (P(D|M2) * P(M2))/P(D).

Calculation

Substituting the given values into Bayes' theorem, we get: P(M2|D) = (0.04 * 0.25)/0.022. After simplifying this, we find P(M2|D) = 0.4545 or approximately 45.45%.