Question

A particular concentration of a chemical found in polluted water has been found to be lethal to 20% of the fish that are exposed to the concentration for 24 hours. Twenty fish are placed in a tank containing this concentration of chemical in water. Find the mean and variance of the number that survive.

Answer 1

mean=16

Variance=3.2

Explanation:

P(lethal)=P(not survive)=0.20

P( not lethal)=P(survive)=1-0.2=0.8

The specified scenario indicates binomial distribution because

i) There are two possible outcomes lethal or not lethal.

ii) Probability of not lethal/survive remains constant on each trail.

iiI) The experiment is repeated fixed number of times n=20.

iv) The trails are independent.

The mean for binomial distribution is

E(x)=np

Here, n=20 and p=0.8

E(x)=20*0.8

E(x)=16

The variance of binomial distribution is

V(x)=npq

q=1-p=1-0.8=0.2

So,

V(x)=20*0.8*0.2

V(x)=3.2

Answer 2

Mean: 16

Variance: 3.2

Explanation:

For each fish, there are only two possible outcomes. Either they survive, or they do not. The probability of a fish surviving is independent from other fishes. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

`E(X) = np`

The variance of the binomial distribution is:

`V(X) = np(1-p)`

A particular concentration of a chemical found in polluted water has been found to be lethal to 20% of the fish that are exposed to the concentration for 24 hours.

This means that 80% survive, so
`p = 0.8`

The mean is:

`E(X) = np = 20*0.8 = 16`

The variance is:

`V(X) = np(1-p) = 20*0.8*0.2 = 3.2`