Question

The probability that a lab specimen contains high levels of contamination is 0.14. a group of 4 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?

Answer 1

This situation has two outcomes: either a specimen has high levels of contamination or not. This two outcomes satisfy what we call a binomial distribution (note "bi" in binomial).

The binomial distribution tells us the probability that a randomly selected sample will have the outcome of success. In this case, we consider having high levels of contamination as the "success" outcome (even though we sure hope it does not happen) while not having it means "failure". The distribution takes the form:

$P(X=x)= _{n}C_{x} p^{x} q^{n-x}$
where n is the total number of samples, x is the number of samples you'll expect to have a success outcome, p is the probability of success, q is the probability of failure, and nCx is the combination of n samples taken x at a time.

For the next step, let's digest the problem to get the needed variables.

The total number of samples, n, is equal to 4. The probability of success (having high levels of contamination), p, is 14% or 0.14, and the probability of failure (not receiving a discount) is 86% or 0.86.

For P(X=x), we need to find the probability that none contain high levels of contamination. This is equivalent to saying
$P(X=0)$

Calculating the probability we'll get:
$P(X=0)= _{4}C_{0} (0.14)^{0} (0.86)^{4}=0.547008$

ANSWER: The probability that none contain high levels of contamination is 0.547008 or 54.7008%