Question

A speaks the truth 75% of the cases and B in 80% of the cases. What is the probability that their statements about an incident do not match?

Answer 1

Hint: To solve this question, we will use the concept of mutually exclusive events. In general, two events A and B are said to be mutually exclusive events if the occurrence of one of them excludes the occurrence of the other event, i.e. if they cannot occur simultaneously.

Complete step-by-step answer:
Given that,
A speaks the truth 75% of the time.
So, the probability of A speaks the truth will be,
$ \Rightarrow P\left( A \right) = \dfrac{{75}}{{100}} = \dfrac{3}{4}$
B speaks truth 80% of the cases,
So, the probability of B speaks the truth will be,
$ \Rightarrow P\left( B \right) = \dfrac{{80}}{{100}} = \dfrac{4}{5}$
We have to find the probability that their statements about an incident do not match.
As we know that,
The probability of event ‘not A’ is given by,
$ \Rightarrow P\left( {notA} \right) = P\left( {A'} \right) = 1 - P\left( A \right)$
So, the probability that A does not speaks the truth will be,
$ \Rightarrow P\left( {A'} \right) = 1 - \dfrac{3}{4} = \dfrac{{4 - 3}}{4}$
$ \Rightarrow P\left( {A'} \right) = \dfrac{1}{4}$
Similarly,
The probability that B does not speaks the truth will be,
$ \Rightarrow P\left( {B'} \right) = 1 - \dfrac{4}{5} = \dfrac{{5 - 4}}{5}$
$ \Rightarrow P\left( {B'} \right) = \dfrac{1}{5}$
Let E be the event their statements about an incident do not match.
There will be two conditions when their statements do not match.
Condition 1: When A speaks truth but B don’t.
Condition 2: When B speaks truth but A don’t.
So,
The probability of event E will be given as,
$ \Rightarrow P\left( E \right) = P\left( A \right)P\left( {B'} \right) + P\left( B \right)P\left( {A'} \right)$
Putting all the values, we will get
$ \Rightarrow P\left( E \right) = \left( {\dfrac{3}{4}} \right)\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{4}{5}} \right)\left( {\dfrac{1}{4}} \right)$
\[ \Rightarrow P\left( E \right) = \dfrac{3}{{20}} + \dfrac{1}{5}\]
Taking L.C.M as 20,
\[ \Rightarrow P\left( E \right) = \dfrac{{3 + 4}}{{20}}\]
\[ \Rightarrow P\left( E \right) = \dfrac{7}{{20}}\]
Hence, the probability that their statements about an incident do not match is \[\dfrac{7}{{20}}\]

Note: Whenever we ask such types of questions, we have to remember some basic concepts of probability. First, we will collect all the given details, then we will use the concept of probability of event ‘not A’. Through this, we will find the other required probabilities. After that, we will make the event according to the question and by putting the values in that event, we will get the required answer.