A cellular phone manufacturer randomly selects 5 phones from every batch of 50 produced. If at least one of the phones is found to be defective than each phone in the batch is tested individually. Find the probability that the entire batch will need testing if the batch contains
a). 8 defective phones
b). 2 defective phones
To find the probability that the entire batch will need testing, we need to consider two scenarios:
Scenario 1: No defective phones are found among the randomly selected phones.
Scenario 2: At least one defective phone is found among the randomly selected phones.
a) If there are 8 defective phones in the batch:
In this case, we need to determine the probability of not finding any defective phones among the 5 randomly selected phones. Since there are 8 defective phones in the batch, the number of non-defective phones in the batch is 50 - 8 = 42.
The probability of not finding any defective phones in the 5 randomly selected phones is:
P(No defective in 5) = (42/50) * (41/49) * (40/48) * (39/47) * (38/46)
If at least one defective phone is found among those 5 phones, the entire batch will need testing. Therefore, the probability that the entire batch will need testing can be calculated as:
P(Batch needs testing with 8 defective) = 1 - P(No defective in 5)
b) If there are 2 defective phones in the batch:
In this case, the number of non-defective phones in the batch is 50 - 2 = 48.
The probability of not finding any defective phones in the 5 randomly selected phones is:
P(No defective in 5) = (48/50) * (47/49) * (46/48) * (45/47) * (44/46)
If at least one defective phone is found among those 5 phones, the entire batch will need testing. Therefore, the probability that the entire batch will need testing can be calculated as:
P(Batch needs testing with 2 defective) = 1 - P(No defective in 5)
By calculating these probabilities, we can find the probabilities that the entire batch will need testing for each scenario.