Two witnesses, A and B, are each asked whether a certain event took place. A speaks the truth in 3 out of 4 cases and B speaks the truth in 4 out of 5 cases, independently of each other. What is the probability that they contradict each other (one says the event took place and the other says it did not)?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\frac{7}{20}\)
There are only two possible answers, so the witnesses contradict each other exactly when one tells the truth and the other does not. If both tell the truth, or both lie, they give the same answer.
P(A true) \(\displaystyle =\frac{3}{4}\), P(A lies) \(\displaystyle =\frac{1}{4}\)
P(B true) \(\displaystyle =\frac{4}{5}\), P(B lies) \(\displaystyle =\frac{1}{5}\)
A true and B lies \(\displaystyle =\frac{3}{4}\times\frac{1}{5}=\frac{3}{20}\)
A lies and B true \(\displaystyle =\frac{1}{4}\times\frac{4}{5}=\frac{4}{20}\)
P(they contradict) \(\displaystyle =\frac{3}{20}+\frac{4}{20}=\frac{7}{20}\)