Question

The probability that a lab specimen contains high levels of contamination is 0.15. A group of 3 independent samples are checked. Round your answers to four decimal places (e.g. 98.7654). (a) What is the probability that none contain high levels of contamination? (b) What is the probability that exactly one contains high levels of contamination? (c) What is the probability that at least one contains high levels of contamination?

Answer 1

Answer:

a) There is a 61,41% of none of the samples containing high levels of contamination.

b)There is a 32.52% probability that exactly one sample contains high levels of contamination.

c) There is a 38.59% probability that at least one contains high levels of contamination

Step-by-step explanation:

The probabilities are independent from each other. It means that the probability of selecting a lab specimen being contaminated is always 15%, no matter how many contaminated lab specimen have been chosen.

a) There are 3 independent samples. For each sample, the probability of it not being contaminated is 85%. So, the probability that none of the sample are contaminated is

$P = (0.85)^3 = 0.6141 = 61,41%$

There is a 61,41% of none of the samples containing high levels of contamination.

b) There are 3 independent samples. For each sample, the probability of it being contaminated is 15% and not contaminated 85%.

So the probability the exactly one sample contains high levels of contamination is:

$P = (0.85)^2(0.15) = 0.1084 = 10,84%$

There can be 3 orderings of the sample in these conditions.(C-NC-NC, NC-C-NC, NC-NC,C), so the probability that exactly one contains high levels of contamination is

P = 3*0.1084 = 0.3252 = 32.52%.

There is a 32.52% probability that exactly one sample contains high levels of contamination.

c) The sum of the probabilities is always 100%.

In relation to the existence of a contaminated sample, either:

-None of the samples are contaminated.

-At least one of the samples are contaminated.

So, the probability of at least one of the samples being contaminated is 100% - the probability that none of the samples are contaminated, that we have already found in a).

So, it is

100% - 61.41% = 38.59%

There is a 38.59% probability that at least one contains high levels of contamination