Question

In a given race, the odds in favor of four horses $A,B,C\& D$are $1:3,1:4,1:5$ and $1:6$respectively. Assuming that a dead heat is impossible, find the chance that one of them wins the race

$$(A) \dfrac{{101}}{{420}} \\\ (B) \dfrac{{101}}{{210}} \\\ (C) \dfrac{{319}}{{420}} \\\ (D) \dfrac{{111}}{{420}} \\\$$

Answer 1

We will use the information of the ratio and by the help of the ratio we will determine the value of
$P(A),P(B),P(C),P(D)$. So, by the given relation of the ratio we will be able to determine them and in the last we can add them to get the required solution.

Complete step-by-step answer:
In the given question it is given that the odds is in favor
Let us consider our first case that is for A
The ratio for A can be represented as follows
$\dfrac{{P(A)}}{{P(\overline {A)} }} = \dfrac{1}{3}$
Since we need to find the value of $P(A)$ we need to know the relation of $P(\overline {A)}$ in terms of $P(A)$ that is $1 - P(A)$ . So, substituting the value in the required place so that we can simplify for the specific value
$\dfrac{{P(A)}}{{1 - P(A)}} = \dfrac{1}{3}$
Simplifying the above for the value of $P(A)$ we get
$3P(A) = 1 - P(A)$
Taking $P(A)$ on one side
$4P(A) = 1$
Hence, we get
$P(A) = \dfrac{1}{4}$
So, similarly we can simplify for $P(B)$
$\dfrac{{P(B)}}{{P(\overline {B)} }} = \dfrac{1}{4}$
Since we need to find the value of $P(B)$ we need to know the relation of $P(\overline {B)}$ in terms of $P(B)$ that is $1 - P(B)$ . So, substituting the value in the required place so that we can simplify for the specific value
$\dfrac{{P(B)}}{{1 - P(B)}} = \dfrac{1}{4}$
Simplifying the above for the value of $P(B)$ we get
$4P(B) = 1 - P(B)$
Taking $P(B)$ on one side
$5P(B) = 1$
Hence, we get
$P(B) = \dfrac{1}{5}$
So, similarly we can simplify for $P(C)$
$\dfrac{{P(C)}}{{P(\overline {C)} }} = \dfrac{1}{5}$
Since we need to find the value of $P(C)$ we need to know the relation of $P(\overline {C)}$ in terms of $P(C)$ that is $1 - P(C)$ . So, substituting the value in the required place so that we can simplify for the specific value
$\dfrac{{P(C)}}{{1 - P(C)}} = \dfrac{1}{5}$
Simplifying the above for the value of $P(C)$ we get
$5P(C) = 1 - P(C)$
Taking $P(C)$ on one side
$6P(C) = 1$
Hence, we get
$P(C) = \dfrac{1}{6}$
So, similarly we can simplify for $P(D)$
$\dfrac{{P(D)}}{{P(\overline {D)} }} = \dfrac{1}{6}$
Since we need to find the value of $P(D)$ we need to know the relation of $P(\overline {D)}$ in terms of $P(D)$ that is $1 - P(D)$ . So, substituting the value in the required place so that we can simplify for the specific value
$\dfrac{{P(D)}}{{1 - P(D)}} = \dfrac{1}{6}$
Simplifying the above for the value of $P(D)$ we get
$6P(D) = 1 - P(D)$
Taking $P(D)$ on one side
$7P(D) = 1$
Hence, we get
$P(D) = \dfrac{1}{7}$
Hence, we get
$P(A) = \dfrac{1}{4}$
$P(B) = \dfrac{1}{5}$
$P(C) = \dfrac{1}{6}$
$P(D) = \dfrac{1}{7}$
The sum of the above
$P(A) + P(B) + P(C) + P(D) = \dfrac{1}{4} + \dfrac{1}{5} + \dfrac{1}{6} + \dfrac{1}{7}$
So, On simplifying the above we get
$P(A) + P(B) + P(C) + P(D) = \dfrac{{319}}{{420}}$

Option C is the correct answer.

Note: Probability of any event is defined as the possible outcomes by the total number of outcomes.
${\text{Probability}}= \dfrac{\text{Possible Outcomes}}{\text{Total number of Outcomes}}$
The probability of any event lies between 0 and 1.