Two machines turn out all the products in a factory, with the first machine producing 45% of the product and the second 55%. The first machine produces defective products 8% of the time and the second machine 9% of the time
What is the probability that a defective part is produced at the factory?
To find the probability that a defective part is produced at the factory, we need to consider the probability of each machine producing a defective part and combine them.
We first find the probability that a defective part is produced by each machine:
- For the first machine, since it produces 45% of the products and 8% of them are defective: 0.45 * 0.08 = 0.036 or 3.6% probability of producing a defective part.
- For the second machine, since it produces 55% of the products and 9% of them are defective: 0.55 * 0.09 = 0.0495 or 4.95% probability of producing a defective part.
To find the overall probability of a defective part being produced, we simply add the probabilities for each machine: 0.036 + 0.0495 = 0.0855 or 8.55% probability that a defective part is produced at the factory.
To find the probability that a defective part is produced at the factory, we can use the concept of conditional probability.
Let's denote the events as follows:
A - Defective part is produced
B1 - Part produced by the first machine
B2 - Part produced by the second machine
We are given the following information:
P(B1) = 45% = 0.45 (probability that the part is produced by the first machine)
P(B2) = 55% = 0.55 (probability that the part is produced by the second machine)
P(A|B1) = 8% = 0.08 (probability of a defective part given that it was produced by the first machine)
P(A|B2) = 9% = 0.09 (probability of a defective part given that it was produced by the second machine)
We want to find P(A), the probability that a defective part is produced at the factory.
Using the law of total probability, we can calculate P(A) as follows:
P(A) = P(A|B1) * P(B1) + P(A|B2) * P(B2)
P(A) = 0.08 * 0.45 + 0.09 * 0.55
P(A) = 0.036 + 0.0495
P(A) = 0.0855
Therefore, the probability that a defective part is produced at the factory is 0.0855, or 8.55%.
To find the probability of a defective part being produced at the factory, we need to consider the probability of each machine producing a defective part and the proportion of products each machine produces.
Let's break down the problem step by step:
Step 1: Find the probability of the first machine producing a defective part. Given that the first machine produces defective products 8% of the time, the probability of the first machine producing a defective part is 0.08.
Step 2: Find the probability of the second machine producing a defective part. Given that the second machine produces defective products 9% of the time, the probability of the second machine producing a defective part is 0.09.
Step 3: Calculate the weighted average of the probabilities. Since the first machine produces 45% of the products and the second machine produces 55% of the products, we can use these proportions as weights to calculate the overall probability of a defective part being produced at the factory.
To calculate the weighted average, we multiply the probability of each event by its corresponding weight and sum them up:
Weighted average = (Probability of the first machine producing a defective part) * (Proportion of products made by the first machine) + (Probability of the second machine producing a defective part) * (Proportion of products made by the second machine)
= (0.08 * 0.45) + (0.09 * 0.55)
= 0.036 + 0.0495
= 0.0855
Therefore, the probability that a defective part is produced at the factory is 0.0855 or 8.55%.