Question

There are 7 horses in a race, If Mr. X places a bet on 2 Horses randomly, what is the probability of Mr. X winning the bet?
A. \[\dfrac{1}{7}\]
B. \[\dfrac{4}{7}\]
C. \[\dfrac{3}{7}\]
D. \[\dfrac{2}{7}\]

Answer 1

Hint: We will find the number of ways of choosing 2 horses out of 7 horses. Now calculate the number of ways that 1 horse will win out of 2 horses. Then apply the probability formula to calculate the required solution.

Formula used:
The probability of an event happening is,
\[P(E) = \dfrac{{{\text{Favourable Outcomes}}}}{{{\text{Total Outcomes}}}}\]
The combination formula: \[^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\]

Complete step-by-step solution:
There are 7 horses in a race.
The Possible ways of randomly choosing 2 horses out of total 7 is given by,\[^7{C_2}\]
Now apply combination formula
\[ = \dfrac{{7!}}{{2! \times (7 - 2)!}}\]
\[ = \dfrac{{7 \times 6 \times 5!}}{{2 \times 1 \times 5!}}\]
Cancel \[5!\] from the denominator and numerator
\[ = \dfrac{{7 \times 6 }}{{2 \times 1 }}\]
\[ = 21\,{\rm{ways}}\]
The possible ways of choosing 2 horses randomly 1 of which is a winner and rest 6 are losing horses is,
\[\begin{array}{l}{\,^1}{C_1} \times {\,^6}{C_1}\\ \Rightarrow 1 \times 6\\ \Rightarrow 6\end{array}\]
Let \[E\] be the event of Mr. X winning the bet.
Then, the probability of Mr. X winning the bet is,
\[P(E) = \dfrac{{{\text{Favourable Outcomes}}}}{{{\text{Total Outcomes}}}}\]
\[ \Rightarrow P(E) = \dfrac{6}{{21}}\]
Divide the numerator and denominator by 3.
\[ \Rightarrow P(E) = \dfrac{2}{7}\]
Hence, the probability of Mr. X winning the bet is \[\dfrac{2}{7}\].
Hence option D is the correct option.

Note: Students often confused with permutation and combination. The combination is the process of counting the choices we make from a set of \[n\] things. The permutation is the process of counting all possible configurations out of \[n\] objects.
In the given question we find the number of selection of horses. So, we used the combination formula.