Question

At the homecoming game at a security checkpoint, 100 persons were randomly selected for extra screening to determine if they were bringing any alcohol, drugs, or weapons into the stadium (i.e., breaking the rules). We know the screening is 98% accurate when testing a person who is breaking the rules and 95% accurate when testing a person who is not breaking the rules. Suppose (if we were "all knowing") that 20 of the tested persons were in fact breaking the rules. What is the probability a particular person is not breaking the rules, given that the screening test indicated that he/she was breaking the rules? Define the appropriate events! Summarize the given information based on the events that you define, then complete all calculations. Clearly indicate/label your final answer.

Answer 1

Define the events

B a person is breaking the rules

screening iS tve = screening is-ve

The following probabilities are defined.

(B) = 20/ 100 = 0.2

P(+B) = 0.98 P(-B) 0.95

Using complementary probability,

P(B) 1 P(B)0.8 P-B) = 1-P(+B) = 1-0.98 = 0.02 P(+B) 1- P-B) 1 0.95 0.05

Using total probability theorem,

P(-) = 0.02(0.2) +0.95(1-02) P(-) = 0.764 P(+) = 1-0.764 = 0.236

The required conditional probability is

0.05 x 0.8 P(Bj +) =-0.236 P(B|+) = 0.1695