Question: A shipment of 30 inexpensive digital watches, including 7 that are defective, is sent to a…

A shipment of 30 inexpensive digital watches, including 7 that are defective, is sent to a department store. The receiving department selects 10 at random for testing and rejects the whole shipment if 1 or more in the sample are found defective. What is the probability that the shipment will be rejected? The probability the shipment will be rejected is (Simplify your answer. Type an integer or decimal rounded to two decimal places as needed.)

Solution

To find the probability that the shipment will be rejected, we need to calculate the probability of selecting at least one defective watch out of the 10 chosen. First, calculate the probability of selecting no defective watches (which is the complement of selecting at least one defective watch). Then subtract that from 1 to find the probability of rejecting the shipment. The total number of watches is 30, with 7 defective and 23 non-defective. The probability of selecting all non-defective watches: \[ \frac{\binom{23}{10}}{\binom{30}{10}} \] Calculate the combinations: \[ \binom{23}{10} = \frac{23!}{10! \times (23-10)!} = 1144066 \] \[ \binom{30}{10} = \frac{30!}{10! \times (30-10)!} = 30045015 \] Now, calculate the probability of selecting no defective watches: \[ \frac{1144066}{30045015} \approx 0.0386 \] Subtract from 1 to find the probability of at least one defective watch: \[ 1 - 0.0386 = 0.9614 \] Thus, the probability that the shipment will be rejected is approximately 0.96.