Question

The probability that $A$ speaks truth is $\dfrac{4}{5}$, while this probability for $B$ is $\dfrac{3}{4}$. The probability that they contradict each other when asked to speak on a fact
A. $\dfrac{4}{5}$
B. $\dfrac{1}{5}$
C. $\dfrac{7}{{20}}$
D. $\dfrac{3}{{20}}$

Answer 1

Here the given question is based on the concept of probability. Given the probability of $A$speaks truth and probability of $B$ speaks truth. We have to find the probability of they contradict each other by using the equation $P\left( A \right) \cdot P\left( {B'} \right) + P\left( {A'} \right) \cdot P\left( B \right)$. Where $P\left( {A'} \right)$ is the probability that $A$ is not speaking the truth and $P\left( {B'} \right)$ is the probability that $B$ is not speaking truth.

Complete step by step answer:
Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e., how likely they are to happen, using it. Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event. The probability formula is defined as the probability of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.
$\text{Probability of event to happen}\,P\left( E \right) = \dfrac{\text{Number of favourable outcomes}}{\text{Total Number of outcomes}}$

Consider the given question: The probability that $A$speaking truth $P\left( A \right) = \dfrac{4}{5}$
The probability that $B$ speaks truth $P\left( B \right) = \dfrac{3}{4}$
Hence, the probability that $A$ is not speaking truth or $A$ speaks lie is:
$\Rightarrow \,\,P\left( {A'} \right) = 1 - P\left( A \right)$
$\Rightarrow \,\,P\left( {A'} \right) = 1 - \dfrac{4}{5}$
Taking 5 as LCM, then
$\Rightarrow \,\,P\left( {A'} \right) = \dfrac{{5 - 4}}{5}$
On simplification, we get
$\Rightarrow \,\,P\left( {A'} \right) = \dfrac{1}{5}$
The probability that $B$ is not speaking truth or $B$ speaks lie is:
$\Rightarrow \,\,P\left( {B'} \right) = 1 - P\left( B \right)$
$\Rightarrow \,\,P\left( {B'} \right) = 1 - \dfrac{3}{4}$
Taking 4 as LCM, then
$\Rightarrow \,\,P\left( {B'} \right) = \dfrac{{4 - 3}}{4}$
On simplification, we get
$\Rightarrow \,\,P\left( {B'} \right) = \dfrac{1}{4}$

Now find the probability that they contradict each other when asked to speak on a fact is:
$P\left( \text{ contradict each other } \right) = P\left( \text{ A speaks truth} \right) \cdot P\left( \text{ B speaks lie} \right) + P\left( \text{A speaks lie} \right) \cdot P\left( \text{B speaks truth} \right)$
$\Rightarrow \,\,\,P\left( A \right) \cdot P\left( {B'} \right) + P\left( {A'} \right) \cdot P\left( B \right)$
On substituting the values, we have
$\Rightarrow \,\,\,\left( {\dfrac{4}{5}} \right) \cdot \left( {\dfrac{1}{4}} \right) + \left( {\dfrac{1}{5}} \right) \cdot \left( {\dfrac{3}{4}} \right)$
$\Rightarrow \,\,\,\dfrac{4}{{20}} + \dfrac{3}{{20}}$
On simplification we get
$\Rightarrow \,\,\,\dfrac{{4 + 3}}{{20}}$
$\therefore \,\,\,\dfrac{7}{{20}}$
Hence, the probability of contradicting the truth of $A$ and $B$ each other is $\dfrac{7}{{20}}$.

Therefore, option C is the correct answer.

Note: The probability is a number of possible values. Candidates must know the basic theorem that is addition and multiplication theorem. Remember the complement of an event is the event which is not occurring. If the probability that Event A will not occur is denoted by $P\left( {A'} \right)$ which is equal to $P\left( {A'} \right) = 1 - P\left( A \right)$.Contradict means opposite Like if A speaks truth than B speaks lie and vice versa.