Question

A speaks truth in $60\% $ cases and B in $90\% $ cases. In what percentage of cases are they likely to contradict each other in stating the same fact?
A). $0.42$
B). $0.58$
C). $0.46$
D). None of these

Answer 1

Hint: Probability is the possibility of occurrence of an event. In order to solve the given problem we need to consider two different situations satisfying the conditions given in the question. The condition is to contradict each other on the same fact. For contradicting each other one has to speak the truth and the other should not speak the truth.

Complete step-by-step solution:
Let $P(A)$ and $P(B)$ be the probability of speaking the truth of A and B respectively. Therefore,
$P(A) = \dfrac{{60}}{{100}} = 0.6$
$P(B) = \dfrac{{90}}{{100}} = 0.9$
The probability of an event occurring $100\% $ is $1$
Therefore,
The probability of A not speaking the truth is $1 - P(A) = 1 - 0.6 = 0.4$
The probability of B not speaking the truth is $1 - P(B) = 1 - 0.9 = 0.1$
In order to satisfy the given condition i.e., for A and B to contradict each other on the same fact there are two cases.
First Case: A speaking the truth and B not speaking the truth
The probability of simultaneous occurrence of two events can be calculated by multiplying the probabilities of the events.
Therefore, the probability of A speaking the truth and B not speaking the truth is $P(A) \times (1 - P(B))$
$ = 0.6 \times (1 - 0.9)$
$ = 0.6 \times (0.1)$
$ = 0.06$
Second Case: A not speaking the truth and B speaking the truth
Therefore, the probability of A not speaking the truth and B speaking the truth is $(1 - P(A)) \times P(B)$
$ = (1 - 0.6) \times 0.9$
$ = 0.4 \times 0.9$
$ = 0.36$
Therefore, the probability of A and B contradicting each other on the same fact is $P(A) \times (1 - P(B)) + (1 - P(A)) \times P(B)$
$ = 0.06 + 0.36$
$ = 0.42$
Percentage of cases they are likely to contradict each other on the same fact are $0.42 \times 100$
$ = 42\% $
Therefore, Option (D) is correct.

Note: In this type problem first we need to find the different cases that satisfy the given condition. Then calculating the probability of those cases and finally adding them gives the required result. In the given problem there are $4$ cases possible. They are $1.$ A speaking truth and B not speaking truth, $2.$ A not speaking truth and B speaking truth, $3.$ Both A and B speaking truth, $4.$ Both A and B not speaking truth. But of them only two cases satisfy the given condition.