Question

Five horses are in a race. Mr. ‘A’ selected two horses randomly and bet on them. The probability that Mr. ’A’ selected the winning horse is
A. $\dfrac{3}{5}$
B. $\dfrac{1}{5}$
C. $\dfrac{2}{5}$
D. $\dfrac{4}{5}$

Answer 1

Hint : When $n$ objects are to be placed in $r$ seats randomly, then this can be done in ${n^r}$ ways. Use the formula ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ in this question to reduce the complication of this question. Also, the number of ways of arranging $n$ things is in $n!$ ways, this formula will help to find out the number of ways to arrange things.

Complete step-by-step answer :
The number of ways to select $r$ objects out of $n$ objects is ${}^n{C_r}$ . The formula for ${}^n{C_r}$ is ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ . It is known that ${}^n{C_1} = n$ .
There are in total 5 horses. Mr. A has selected 2 horses randomly.
He can select 2 horses in ${}^5{C_2} = 10$ ways.
We have to find the probability that Mr. A selected the winning horse.
Let us assume that Mr. A has selected the winning horse. He needs to select one more horse.
He can select one more horse from the 4 remaining horses in ${}^4{C_1}$ ways.
The favourable case is ${}^4{C_1}$ .
The probability is calculated by using the formula ${\rm{Probability = }}\dfrac{{{\text{Number of favorable cases}}}}{{{\text{Total number of cases}}}}$ .
The probability that Mr. ’A’ selected the winning horse is given by $\dfrac{{{}^4{C_1}}}{{10}}$ .
$\Rightarrow \dfrac{{{}^4{C_1}}}{{10}} = \dfrac{4}{{10}}\\\ = \dfrac{2}{5}$

The probability that Mr. ’A’ selected the winning horse is given by $\dfrac{2}{5}$.

So, the correct answer is “Option C”.

Note : Students must avoid mistakes while using the formula for calculating The number of ways to select $r$ objects out of $n$ objects is ${}^n{C_r}$ . Instead of taking the number of ways as ${}^n{C_r}$ , students can take it mistakenly as ${}^r{C_n}$ . Also, the language of the question should be understood in a clear manner so that conditions are well expressed in terms of mathematical symbols.