Answer:
a) 95.29% probability that the test comes back negative for all six people
b) 4.71% probability that the test comes back positive for at least one of the six people
Step-by-step explanation:
For each person who do not have the antibody and are given the test, there are only two possible outcomes. Either the test comes back negative, or it comes back positive. The probability of the test coming back negative for a person is independent of any other person. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
99.2% effective
So [tex]p = 0.992[/tex]
six randomly selected people
This means that [tex]n = 6[/tex]
(a) What is the probability that the test comes back negative for all six people?
This is P(X = 6).
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 6) = C_{6,6}.(0.992)^{6}.(0.008)^{0} = 0.9529[/tex]
95.29% probability that the test comes back negative for all six people
(b) What is the probability that the test comes back positive for at least one of the six people?
Either it comes negative for all six people, or it comes positive for at least one of them. The sum of the probabilities of these events is 100%. So
p + 95.29 = 100
p = 4.71
4.71% probability that the test comes back positive for at least one of the six people
