Question

Out of 15 persons 10 can speak Hindi and 8 can speak English. If two persons are chosen at random, then the probability that one person speaks Hindi only and the other speaks both Hindi and English is
$
  \left( a \right)\dfrac{5}{3} \\
  \left( b \right)\dfrac{7}{{12}} \\
  \left( c \right)\dfrac{1}{5} \\
  \left( d \right)\dfrac{2}{5} \\
 $

Answer 1

Hint: In this question, we have to use the property of Sets as well as the concept of probability. We have to use $n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n(A \cap B)$ and Probability of an event happening $ = \dfrac{{{\text{Number of ways it can happen}}}}{{{\text{Total number of outcomes}}}}$.

Complete step-by-step answer:

Total number of persons $n\left( {H \cup E} \right) = 15$
Number of persons who speak Hindi $n\left( H \right) = 10$
Number of persons who speak English $n\left( E \right) = 8$
Now, we have to find a number of people who speak both Hindi and English.
So, we use the property of sets $n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n(A \cap B)$ .
Number of persons who speak both Hindi and English $n\left( {H \cap E} \right)$ .
$
   \Rightarrow n\left( {H \cup E} \right) = n\left( H \right) + n\left( E \right) - n(H \cap E) \\
   \Rightarrow 15 = 10 + 8 - n(H \cap E) \\
   \Rightarrow n(H \cap E) = 18 - 15 \\
   \Rightarrow n(H \cap E) = 3 \\
 $
Number of persons who speak Hindi only $ = n\left( H \right) - n\left( {H \cap E} \right) = 10 - 3 = 7$
Total ways of selecting 2 people, one person speaks Hindi only and the other speaks both Hindi and English $ = {}^{15}{C_2}$
Using ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
$
   \Rightarrow {}^{15}{C_2} = \dfrac{{15!}}{{\left( {2!} \right) \times \left( {13!} \right)}} \\
   \Rightarrow \dfrac{{15 \times 14 \times 13!}}{{2 \times 13!}} = 105 \\
$
Favourable ways, one person speaks Hindi only and the other speaks both Hindi and English $ = {}^7{C_1} \times {}^3{C_1}$
$
   \Rightarrow \dfrac{{7!}}{{6!}} \times \dfrac{{3!}}{{2!}} \\
   \Rightarrow 7 \times 3 = 21 \\
 $
Probability that one person speaks Hindi only and the other speaks both Hindi and English $ = \dfrac{{{\text{Favourable ways}}}}{{{\text{Total ways}}}}$
\[ \Rightarrow P({\text{Event) = }}\dfrac{{21}}{{105}} = \dfrac{1}{5}\]
So, the correct option is (c).

Note: Whenever we face such types of problems we use some important points. First we find the number of persons who speak both languages by using the property of sets then find total and favourable ways with the help of formula ${}^n{C_r}$ .
So, after using the probability formula we will get the required answer.