New test is 95% negative on patients who do not have the disease. If test is used in 8 healthy volunteers, what is the probability of getting at least one positive result?

a) 1- 0.05 X 8
b) 0.05 X 8
c) 0.05 ^ 8
d) 0.95 ^ 8
e) 1 - 0.95 ^ 8

^ means degree (like 2^3=8)

The answer is E) 1 - 0.95 ^ 8

Explanation:

Short version

P(+) = p = 0.05
P(-) = q = 0.95
n = 8

The probability of getting AT LEAST one positive = P(X>=1)
This is the same as:

1 - P(X=0) = 1 - 0.95^8

P(X=0) = always p^n

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Long version

This follows a Binomial distribution, where:

P(X=0) = nCr * p^r * (q)^(n-r)
P(X=0) = (n combinations r) * p^r * (q)^(n-r)
P(X=0) = 8C0 * 0.05^0 * (0.95)^(8-0)
P(X=0) = 1 * 0.95^8 = 0.95^8