A manufacturer makes two models of an item: model I, which accounts for 80% of unit sales, and model II which accounts for 20% of unit sales. Because of defects, the manufacturer has to replace (or exchange) 10% of its model I and 18% of model II. If a model is selected at random, find the probability that it will be defective.
To find the probability that a randomly selected model will be defective, we need to calculate the overall probability of a defect, taking into account the different proportions of the two models and the defect rates for each model.
Let's break it down step by step:
1. Calculate the probability of selecting Model I: Since Model I accounts for 80% of unit sales, the probability of selecting Model I is 0.8.
2. Calculate the probability of selecting Model II: Since Model II accounts for 20% of unit sales, the probability of selecting Model II is 0.2.
3. Calculate the probability of a defect in Model I: As stated, 10% of Model I units are defective. So the probability of a defect in Model I is 0.1.
4. Calculate the probability of a defect in Model II: Similarly, 18% of Model II units are defective. So the probability of a defect in Model II is 0.18.
5. Calculate the overall probability of a defective unit: To calculate the overall probability, we need to consider the probability of selecting each model and the probability of a defect in each model. We can use the law of total probability: P(defective) = P(Model I) * P(defect in Model I) + P(Model II) * P(defect in Model II). Substitute the respective values into the formula:
P(defective) = 0.8 * 0.1 + 0.2 * 0.18
Simplifying this expression:
P(defective) = 0.08 + 0.036
P(defective) = 0.116
Therefore, the probability that a randomly selected model will be defective is 0.116 or 11.6%.