In a batch of 8,000 clock radios, 9% are defective. A sample of 8 clock radios is randomly selected without replacement from the 8,000 and tested. What is the probability that the... In a batch of 8,000 clock radios, 9% are defective. A sample of 8 clock radios is randomly selected without replacement from the 8,000 and tested. What is the probability that the entire batch will be rejected if at least one of those tested is defective?
Understand the Problem
The question involves calculating the probability that at least one out of eight randomly selected clock radios is defective, given that 9% of a batch of 8,000 radios are defective. If at least one is defective, the entire batch will be rejected. We need to determine this probability.
Answer
The probability that the entire batch will be rejected is approximately $0.4868$.
Answer for screen readers
The probability that the entire batch will be rejected is approximately $0.4868$.
Steps to Solve
- Determine the Probability of a Radio Being Defective
Given that 9% of the radios are defective, we calculate the probability of selecting a defective radio. This is represented as:
$$ P(\text{defective}) = 0.09 $$
- Determine the Probability of a Radio Being Non-defective
The probability of selecting a non-defective radio is the complement of the defective probability:
$$ P(\text{non-defective}) = 1 - P(\text{defective}) = 1 - 0.09 = 0.91 $$
- Calculate the Probability of All Eight Radios Being Non-defective
To find the probability that all selected radios are non-defective, we raise the probability of selecting one non-defective radio to the power of the number of radios selected (8):
$$ P(\text{all non-defective}) = (P(\text{non-defective}))^8 = (0.91)^8 $$
- Calculate the Probability of At Least One Defective Radio
The probability of at least one defective radio is the complement of the probability that all are non-defective:
$$ P(\text{at least one defective}) = 1 - P(\text{all non-defective}) $$
- Perform the Calculation
Calculating the value step-by-step:
First, compute ( (0.91)^8 ):
$$ (0.91)^8 \approx 0.5132 $$
Then, calculating the final probability:
$$ P(\text{at least one defective}) = 1 - 0.5132 \approx 0.4868 $$
The probability that the entire batch will be rejected is approximately $0.4868$.
More Information
This probability indicates there's nearly a 49% chance that at least one radio out of eight will be defective, leading to the rejection of the entire batch of 8,000 radios.
Tips
- Confusing the probabilities of defective and non-defective radios.
- Not using the complement rule correctly to find the probability of at least one defective radio.
- Miscalculating ( (0.91)^8 ).
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