Question

A speaks the truth in 60% of the cases, and B in 90% of the cases. In what percentage of cases are they likely to state the same fact?

Answer 1

Hint: Probability ( an event ) $ = \dfrac{{{\rm{\text{number of favourable outcomes}}}}}{{{\rm{\text{total number of outcomes}}}}}$
 Same fact would be given by them only if both speak the truth or both do not speak the truth.
Percentage could be converted into probability and probability has a total value of 1.
If probability of occurring of something is p then probability of not occurring would be 1-p.

Complete step by step solution:
Step 1
Calculate the probability of A and B speaking truth
\[\begin{array}{l}
{\rm{\text{Probability of A speaking truth = }}}{{\rm{P}}_A} = \dfrac{{60}}{{100}} = \dfrac{6}{{10}}\\
{\rm{\text{Probability of B speaking truth =} }}{{\rm{P}}_B} = \dfrac{{90}}{{100}} = \dfrac{9}{{10}}
\end{array}\]
Here, $P_A$ and $P_B$ come out to be 6/10 and 9/10.

Step 2
Calculate the probability of A and B not speaking truth\[\begin{array}{l}
{\rm{\text{Probability of A not speaking truth = }}}{{\rm{Q}}_{{\rm{A}}'}} = 1 - {{\rm{P}}_A} = 1 - \dfrac{6}{{10}} = \dfrac{4}{{10}}\\
{\rm{\text{Probability of B not speaking truth = }}}{{\rm{Q}}_{{\rm{B}}'}} = 1 - {{\rm{P}}_B} = 1 - \dfrac{9}{{10}} = \dfrac{1}{{10}}
\end{array}\]

Step 3
They are likely to provide the same fact when both speaking truth and both not speaking truth.

Step 4
$\begin{array}{l}
\therefore {\rm{\text{Probability(both states the same fact) = Probability(both A and B tell the truth)}}}\\
{\rm{ + }}\\
{\rm{\text{Probability(both A and B lie)}}}
\end{array}$
     \[{{\rm{P}}_{\rm{s}}}{\rm{ = }}{{\rm{P}}_{\rm{A}}}{\rm{ \times }}{{\rm{P}}_{\rm{B}}}{\rm{ + }}{{\rm{Q}}_{\rm{A}}}{\rm{ \times }}{{\rm{Q}}_B}\]

Substitute the values and find
\[\begin{array}{l}
{{\rm{P}}_{\rm{S}}} = \dfrac{6}{{10}} \times \dfrac{9}{{10}} + {\rm{ }}\dfrac{4}{{10}} \times \dfrac{1}{{10}}\\
{{\rm{P}}_{\rm{S}}} = {\rm{ }}\dfrac{{54}}{{100}} + \dfrac{4}{{100}} = \dfrac{{58}}{{100}}
\end{array}\]
Here, PS is the probability of stating the same fact which comes out to be 58/100.

Step 5
Express the probability in percentage
\[{\rm{\text{Percentage of}}}\dfrac{{58}}{{100}} = \dfrac{{58}}{{100}} \times 100 = 58\% \]
The percentage comes out to be 58%.

Therefore, In 58% of the cases A and B, are likely to state the same fact.

Note:
Probability of any event P(A), is always “greater than and equal to 0” and “less than and equal to 1”.
 $0 \le {\rm{P(A)}} \le {\rm{1}}$.
‘AND’ operations are substituted with $' \times '$, implying both are likely to happen together.
‘OR’ operations are substituted with $' + '$, implying either of the one possibility is likely to happen.