Question

red snapper is a rare and expensive reef fish served at upscale restaurants. a certain law prohibits restaurants from serving a cheaper, look - alike variety of fish (vermilion snapper or lane snapper) to customers who order red snapper. researchers at a university used dna analysis to examine fish specimens labeled
ed snapper\ that were purchased from vendors across the country. the dna tests revealed that 70% of the specimens were not red snapper, but the cheaper, look - alike variety of fish. a. assuming that the results of the dna analysis are valid, what is the probability that you are actually served red snapper the next time you order it at a restaurant? the probability is 0.3. (round to the nearest hundredth as needed.) b. if there are seven customers at a restaurant, all who have ordered red snapper, what is the probability that at least one customer is actually served red snapper? the probability is (round to four decimal places as needed.)

Explanation:

Step1: Find the probability of not being served red snapper

The probability that a specimen labeled "red snapper" is not red - snapper is $1 - 0.3=0.7$.

Step2: For 7 customers, find the probability that none of them is served red snapper

If we assume the orders of different customers are independent events, the probability that none of the 7 customers is served red snapper is $(0.7)^7$.
\[P(\text{none})=(0.7)^7 = 0.0823543\]

Step3: Find the probability that at least one customer is served red snapper

The probability that at least one customer is served red snapper is the complement of the event that none of them is served red snapper. Let $P(X\geq1)$ be the probability that at least one customer is served red snapper. Then $P(X\geq1)=1 - P(X = 0)$.
\[P(X\geq1)=1-(0.7)^7=1 - 0.0823543 = 0.9176457\approx0.9176\]

Answer:

$0.9176$