A product is assembled using four components. Each component has a 20% probability of being defective. What is the probability that the assembled product is also defective?
each component has an 80% chance of being good
so the probability of a good (not defective) product is ... .8^4
probability of a defective product is
... 1 - .8^4
To find the probability that the assembled product is also defective, we need to consider all possible combinations of defectiveness among the four components.
Since each component has a 20% (0.2) probability of being defective, the probability of a component being non-defective is 1 - 0.2 = 0.8.
Let's consider the possibilities for the product to be defective:
1. All four components are defective:
The probability of a single component being defective is 0.2, so the probability of all four components being defective is 0.2 * 0.2 * 0.2 * 0.2 = 0.0016.
2. Exactly three components are defective:
There are four ways to choose which component is non-defective (since we have four components), and for each of those choices, the probability is 0.8 * 0.2 * 0.2 * 0.2 = 0.0064.
3. Exactly two components are defective:
There are six ways to choose which two components are non-defective (since we have four components), and for each of those choices, the probability is 0.8 * 0.8 * 0.2 * 0.2 = 0.0512.
4. Exactly one component is defective:
There are four ways to choose which component is defective (since we have four components), and for each of those choices, the probability is 0.8 * 0.8 * 0.8 * 0.2 = 0.1024.
To find the probability that the assembled product is defective, we add up the probabilities from all these cases:
0.0016 + 0.0064 + 0.0512 + 0.1024 = 0.1616.
Therefore, the probability that the assembled product is defective is 0.1616, or 16.16%.