Question

Amar, Bimal and Chetan are three contestants for an election, odds against Amar will win is 4 : 1 and odds against Bimal will win is 5 : 1 and odds in favor of Chetan will win 2 : 3 then what is probability that either Amar or Bimal or Chetan will win the election.
A. $\dfrac{23}{20}$
B. $\dfrac{11}{30}$
C. $\dfrac{7}{10} $
D. None of these

Answer 1

Hint:
We have to calculate the probability value using the formula of odd against any event. Here, the value of odd against each case is already given. By doing the necessary substitution calculate the probability. After that check the options that match the values. That will be our final answer.

Formula used:
Odd against $E =\dfrac{1-P\lgroup~E\rgroup}{P\lgroup~E\rgroup}$
where, $1-P\lgroup~E\rgroup$ means event does not occur and $P\lgroup~E\rgroup$ means event occurs.

Complete step-by-step solution:
Given that
4 to 1 are the odds against Amar winning.
This ratio means that while considering the total contestants as 5, the number of persons voted against is 4
Then the votes favor to Amar is 1 out of 5
Similarly, we can write the probabilities of each contestant.
The formula for odds against any event E is
Odd against $E =\dfrac{1-P\lgroup~E\rgroup}{P\lgroup~E\rgroup}$
where, $1-P\lgroup~E\rgroup$ means event does not occur and $P\lgroup~E\rgroup$ means event occurs.
In the case of Amar,
$\dfrac{1-P\lgroup~A\rgroup}{P\lgroup~A\rgroup}=\dfrac{4}{1}$
$\Rightarrow1-P\lgroup~A\rgroup=4P\lgroup~A\rgroup$
$\Rightarrow5P\lgroup~A\rgroup=1$
$\Rightarrow P\lgroup~A\rgroup=\dfrac{1}{5}$
Amar's chances of winning are $P\lgroup~A\rgroup=\dfrac{1}{5}$.
Similarly,
Bimal's chances of winning are $P\lgroup~B\rgroup=\dfrac{1}{6}$.
Chetan's chances of winning are $P\lgroup~C\rgroup=\dfrac{2}{5}$.
As these occurrences are incompatible.
Which means these values don't satisfy the condition. And the options do not match the solutions. So the correct answer will be none of these
So option (d) is the correct answer

Note: By simply dividing the favorable number of possibilities by the entire number of possible outcomes, the probability of an occurrence can be determined using the probability formula. Because the favorable number of outcomes can never exceed the entire number of outcomes, the chance of an event occurring might range from 0 to 1. Additionally, the proportion of positive outcomes cannot be negative.