Question

A speaks truth in 75% of cases and B in 90% of cases. In what percent of cases are they likely to contradict each other in stating the same fact?

Answer 1

To solve this problem, we should know the concept of probability. Let us consider the statement A speaks truth in 75%. We can infer from this that if we ask A a 100 number of cases, A speaks truth for 75 cases and speaks false for 25 cases. We can write using the probability concept that
$P\left( \text{A speaking truth} \right)=\dfrac{75}{100},P\left( \text{A speaking false} \right)=\dfrac{25}{100}$
Likewise we can write the probabilities for B. We can write the probability of two independent events E and F occurring simultaneously by multiplying their probabilities. Mathematically, $P\left( E\text{ and F} \right)=P\left( E \right)\times P\left( F \right)$. We can obtain the required probability of contradicting each other by multiplying the probabilities of A speaking truth and B speaking false or B speaking truth and A speaking false. In the above statement, or can be replaced by + sign when writing probabilities and ’and’ can be replaced by $\times$ sign.

Complete step-by-step answer:
Probability of an event E is defined as the ratio of favourable cases for the event E to the total number of cases. The set of total numbers of cases is called a Sample space. For example, if an event E has $n\left( E \right)$ favourable cases and the sample space has $n\left( S \right)$ elements in it, then the probability of event E is denoted by $P\left( E \right)=\dfrac{n\left( E \right)}{n\left( S \right)}$.
Let us consider the statement A speaks truth in 75%. We can infer from this that if we ask A a 100 number of cases, A speaks truth for 75 cases and speaks false for 25 cases. We can write using the probability concept that
$\begin{aligned} & P\left( \text{A speaking truth} \right)=\dfrac{75}{100}=\dfrac{3}{4} \\\ & P\left( \text{A speaking false} \right)=\dfrac{25}{100}=\dfrac{1}{4} \\\ \end{aligned}$
Let us consider the statement B speaks truthfully 90%. We can infer from this that if we ask B a 100 number of cases, A speaks truth for 90 cases and speaks false for 10 cases. We can write using the probability concept that
$\begin{aligned} & P\left( \text{B speaking truth} \right)=\dfrac{90}{100}=\dfrac{9}{10} \\\ & P\left( \text{B speaking false} \right)=\dfrac{10}{100}=\dfrac{1}{10} \\\ \end{aligned}$
We can write the probability of two independent events E and F occurring simultaneously by multiplying their probabilities. Mathematically,
$P\left( E\text{ and F} \right)=P\left( E \right)\times P\left( F \right)$.
We can understand from the statement persons A and B contradicting each other means that they should not give the same statement for a case. It means that when A is speaking truth, B should speak false and when A is speaking false, B should speak truth. We know that A and B give their statements independently. From the above property, we can write that
$\begin{aligned} & P\left( \text{A speaking truth and B speaking false} \right)=\dfrac{3}{4}\times \dfrac{1}{10}=\dfrac{3}{40} \\\ & P\left( \text{A speaking false and B speaking truth} \right)=\dfrac{1}{4}\times \dfrac{9}{10}=\dfrac{9}{40} \\\ \end{aligned}$
Now, we can infer that the required event occurs for both of the above mentioned scenarios. So, either A speaking truth and B speaking false or A speaking false and B speaking truth results in a case of contradicting each other. So, the total required probability is the sum of them.
$P\left( \text{Contradicting each other} \right)=\dfrac{3}{40}+\dfrac{9}{40}=\dfrac{12}{40}=\dfrac{3}{10}=\dfrac{30}{100}$
From the above final probability, we can infer that A and B contradict each other in 30 number of cases out of 100 cases which means that the percent of cases contradicting each other is 30%.
$\therefore$ A and B are likely to contradict each other in 30% of the cases when stating a fact.

Note: Students can make a mistake if they don’t know the fundamental concept of probability when two events are occurring. The key concept is that, if the two independent events are occurring simultaneously and an ‘and’ condition is applied to get the required statement, we should multiply their individual properties to get the required probability. Likewise, if two independent events occur with a condition that either of them satisfies the given condition, then we have to add their probabilities.