Suppose that 8% of a certain batch of calculators have a defective case, and that 11% have defective batteries. Also, 3% have both a defective case and defective batteries. A calculator is selected from the batch at random. Find the probability that the calculator has a good case and good batteries.
To find the probability that the calculator has both a good case and good batteries, we need to subtract the probability of having either a defective case or defective batteries or both from 1.
Let's calculate step by step:
1. Calculate the probability of having a defective case:
- 8% of calculators have a defective case, so the probability of having a defective case is 8% or 0.08.
2. Calculate the probability of having defective batteries:
- 11% of calculators have defective batteries, so the probability of having defective batteries is 11% or 0.11.
3. Calculate the probability of having both a defective case and defective batteries:
- 3% of calculators have both defects, so the probability of having both defects is 3% or 0.03.
4. Calculate the probability of having either a defective case or defective batteries or both:
- To find this, we add the probability of having a defective case to the probability of having defective batteries and then subtract the probability of having both defects to avoid double counting:
P(defective case or defective batteries) = P(defective case) + P(defective batteries) - P(defective case and defective batteries)
= 0.08 + 0.11 - 0.03
= 0.16.
5. Finally, calculate the probability of having a good case and good batteries:
- We subtract the probability of having either a defective case or defective batteries or both from 1:
P(good case and good batteries) = 1 - P(defective case or defective batteries)
= 1 - 0.16
= 0.84.
Therefore, the probability that the selected calculator has a good case and good batteries is 0.84, or 84%.