Question

Caviar is made from the eggs of sturgeon fish. There are several species of sturgeon. Some are endangered and cannot be harvested for making caviar. Shipments of caviar are DNA tested to make sure they do not contain an illegal species. However, under the de minimus principle of law (minimal violation) a small percentage of illegal species is disregarded. Suppose the percentage of illegal species allowed is 5%. In other words, in a barely allowable shipment, the probability that a randomly chosen egg will be of an illegal species is P(illegal) = 0.05.

a. If six eggs are tested from a shipment of caviar that contains 5% illegal species, calculate the probabilities that 0, 1, 2, 3, 4, 5, 6 of the eggs will be of the illegal species. Give the probabilities to four decimal places.

b. If in fact one egg is illegal, would this be reasonable grounds for suspecting that the percentage of illegal species is greater than 5%? Explain your answer on probabilistic grounds

Answer 1

The probability that a randomly chosen egg will be of an illegal species is Binomial.

The Binomial probability mass function is

Here

a)The probabilities that 0, 1, 2, 3, 4, 5, 6 of the eggs will be of the illegal species are given below. R code also given.

n <- 8
p <- 0.05
x <- 0:n
P <- dbinom (x,n, p)
tab <- data.frame("Number_of_Illegal_Eggs"=x, "Probability"= P)
tab

Number_of_Illegal_Eggs Probability
1 0 6.634204e-01
2 1 2.793349e-01
3 2 5.145643e-02
4 3 5.416467e-03
5 4 3.563465e-04
6 5 1.500406e-05
7 6 3.948438e-07
8 7 5.937500e-09
9 8 3.906250e-11

b) From the above listing of probabilities, the maximum probability occurs when . If increases , this value increases from 0 upto n/2.Hence getting   is a reasonable grounds for suspecting that the percentage of illegal species is greater than 5%.