Question

Red snapper is a rare 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. 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 restaurant, all who have ordered red snapper, what is the probability that at least one customer is actually served red snapper?

Step 1

There are only two possible outcomes when you order red snapper - you either get red snapper or the cheaper look-alike.

Let A be the event that red-snapper is served.

Let B be the event that the cheaper look-alike is served.

Both the outcomes are mutually exclusive ie they can't occur simultaneously. ie P(AnB)= 0

DNA analysis says that 77% of the fish specimens labelled red snapper were not actually red snapper but the cheaper look-alike. So the probability of getting the cheaper look-alike of red snapper on ordering red snapper is 0.77.

This can be written as, P(B) = 0.77

a)

As there are only two mutually exclusive outcomes, we know that P(A) + P(B) = 1

P(A) = 1 - P(B) = 1 - 0.77 = 0.23

Therefore, probability of getting red snapper is 0.23

Step 2

b)

There can be 6 possible outcomes when 5 customers order red snapper.

P(0) = no customer gets red snapper

P(1)= exactly 1 customer gets red snapper

P(2)= exactly 2 customers get red snapper

P(3)= exactly 3 customers get red snapper

P(4)= exactly 4 customers get red snapper

P(5)= All 5 customers get red snapper

As there are just 6 outcomes, P(0) + P(1) + P(2) + P(3) + P(4) + P(5) = 1 .......................equation 1

When there are 5 people i...

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