Question

Are you really being served red snapper? Red snapper is arare and expensive reef fish served at upscale restaurants. Federal law prohibits restaurants from serving a cheaper, look-alike variety of fish (e.g., vermillion snapper or lane snapper) to customers who order red snapper. Researchers

at the University of North Carolina used DNA analysis to examine fish specimens labeled “red snapper” that were purchased from vendors across the country (Nature, July 15, 2004). The DNA tests revealed that 77% of the specimens were not red snapper but the cheaper, look-alike variety of fish.

a. Assuming 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?

b. If there are five customers at a chain restaurant, all who have ordered red snapper, what is the probability that at least one customer is actually served red snapper?

Short answer

(a) The customer served red snapper is 0.23

(b) The probability that at least one customer is actually served red snapper is 0.729

Step-by-step solution

Given information and definitions.

According to the information

A=Specimen is not red snapper.

B=All five specimen is not red snapper.

The formula for probability$\mathbf{P}\mathbf{=}\frac{\mathbf{favourable}\mathbf{}\mathbf{out}\mathbf{}\mathbf{comes}}{\mathbf{total}\mathbf{}\mathbf{out}\mathbf{}\mathbf{comes}}$.

The general rule of compliment is$\mathbf{P}\left(A^c\right)\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{P}\left(A\right)$

Find the probability

$\begin{array}{l}P\left(A\right)=77\%=077\\ P\left(A^c\right)=1-P\left(A\right)\\ =1-0.77\\ =0.23\end{array}$

Therefore,Red snapper represents 0.23 of the customer served fish.

Step 3:The probability that at least one customer is actually served red snapper

$$\begin{gathered} P\left(A⌒B\right)P\left(AandB\right)=P\left(A\right).P\left(B\right) \\ P\left(A\right)={\left(P\left(B\right)\right)}^5 \\ ={\left(0.77\right)}^5 \\ =0.2707 \end{gathered}$$

The complement is

$$\begin{gathered} P\left(B^c\right)=1-P\left(B\right) \\ =1-0.2707 \\ =0.729 \end{gathered}$$

Hence, Red snapper will likely be provided to at least one consumer in 0.729 of cases.