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.