A shipment of portable radios contains 14 good radios and 3 defective ones. If two are randomly selected (without replacement) and tested, find the probability that at least one will be defective.
dtcyd
To find the probability that at least one of the two radios will be defective, we need to calculate the probability of two scenarios: either the first radio is defective and the second is good, or the first radio is good and the second is defective. We can calculate the probability of each scenario and then add them together.
Let's calculate the probability of the first radio being defective and the second being good:
- First, we have 3 defective radios out of a total of 17 radios (14 good + 3 defective).
- Therefore, the probability of selecting a defective radio first is 3/17.
- After selecting the defective radio, there will be 16 radios left, with 14 good radios and 2 defective ones.
- So, the probability of selecting a good radio second is 14/16.
To calculate the probability of the first radio being good and the second being defective, we swap the probabilities:
- The probability of selecting a good radio first is 14/17.
- After selecting the good radio, there will be 16 radios left, with 3 defective ones and 13 good ones.
- So, the probability of selecting a defective radio second is 3/16.
Now, we can calculate the probability of at least one defective radio by adding these two probabilities:
(3/17) * (14/16) + (14/17) * (3/16)
= 42/272 + 42/272
= 84/272
= 0.309
Therefore, the probability that at least one of the two radios selected will be defective is approximately 0.309, or 30.9%.