Question

A speaks the truth 3 out of 4 times, and B 5 out of 6 times. What is the probability that they will contradict each other in stating the same fact?

Answer 1

Hint: To solve this question, we need to know the concept of the probability, we know that the probability is nothing but the ratio of the favorable number of outcomes to the total number of outcomes. Mathematically, we can write it as,
\[\text{Probability = }\dfrac{\text{Favorable Outcomes}}{\text{Total number of outcomes}}\]

Complete step-by-step answer:
In this question, we are asked to find the probability of A and B contradicting each other while stating the same fact.
\[\text{Probability of an event = }\dfrac{\text{Favorable Outcomes}}{\text{Total number of outcomes}}\]
In this question, we are talking about the probability of speaking the truth. So, the number of times stating the true fact to the total number of times stating fact will give us the probability of speaking the truth.
So, we can say,
Probability of A speaking the truth, \[\text{P}\left( A \right)=\dfrac{3}{4}\]
Probability of A not speaking the truth, \[P\left( \overline{A} \right)=\dfrac{1}{4}\]
Probability of B speaking the truth, \[P\left( B \right)=\dfrac{5}{6}\]
Probability of B speaking not speaking the truth, \[P\left( \overline{B} \right)=\dfrac{1}{6}\]
Now, in this question, we have to find the probability of A and B contradicting to each other while stating the same fact. Now, let us consider A speaks the truth but B do not. Then the probability of A and B contradicting is
\[P\left( A\overline{B} \right)=\dfrac{3}{4}\times \dfrac{1}{6}\]
\[=\dfrac{3}{24}\]
\[=\dfrac{1}{8}\]
Now, we will consider the case when B speaks the truth but A does not. So, the probability of A and B contradicting is
\[P\left( \overline{A}B \right)=\dfrac{1}{4}\times \dfrac{5}{6}=\dfrac{5}{24}\]
Now, to find the total probability of A and B contradicting each other, we will sum up the previous two probabilities, that is
Probability of A and B contradicting each other = \[P\left( A\overline{B} \right)+P\left( \overline{A}B \right)\]
\[=\dfrac{1}{8}+\dfrac{5}{24}\]
\[=\dfrac{3+5}{24}\]
\[=\dfrac{8}{24}\]
\[=\dfrac{1}{3}\]
Hence, we can say that the probability of A and B contradicting each other while stating the same fact is \[\dfrac{1}{3}\].

Note: The possible mistake is by finding any one of \[P\left( A\overline{B} \right)\] and \[P\left( \overline{A}B \right)\] and then multiplying it by 2, which is wrong because the probability of A speaking the truth and probability of B speaking the truth are different to each other.