Question

If the odds against winning a race of three horses are respectively 3:1, 4:1 and 5:1, what is the probability that one of these horses will win?

Answer 1

Hint: Let us name the first horse as “a”, second horse as “b” and third horse as “c”. The ratio of odds against winning the horse meaning the ratio of unfavorable outcomes to the favorable outcomes so for horse “a” ratio is 3:1 so multiplying this ratio by x we get unfavorable outcomes as 3x and favorable outcomes as x. The total outcomes will be 4x. We are going to find the probability of winning this horse by taking the ratio of favorable outcomes to the total outcomes so for horse “a” the probability is $ \dfrac{x}{4x} $ . Similarly, find the probability of winning the race for horse “b and c”. Now, to find the probability that one of these horses will win is the addition of all the three probabilities of winning the three horses “a, b and c”.

Complete step-by-step answer:
Let us name the first horse as “a”, second horse as “b” and third horse as “c”.
The ratio of odds against winning for the three horses “a, b and c” are:
The ratio of odds against winning for horse “a” is 3:1
The ratio of odds against winning for horse “b” is 4:1
The ratio of odds against winning for horse “c” is 5:1
Now, we are going to find the probability of winning each horse.
For horse “a”, the ratio of odds against winning the race is the ratio of unfavorable outcomes to the favorable outcomes.
Odd against the winning the race $ =\dfrac{\text{Unfavorable outcomes}}{\text{Favorable outcomes}} $
To convert the ratio to number we are multiplying the ratio by x so the unfavorable outcomes for horse “a” is 3x and favorable outcomes are x
The probability of winning the race by horse “a” is equal to the ratio of favorable outcomes to the total outcomes. Total outcomes we will calculate by adding favorable and unfavorable outcomes so the total probability for horse “a” is 4x.
Probability of winning the race by horse “a” $ =\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}} $
Substituting the value of favorable and total outcomes for horse “a” we get,
Probability of winning the race by horse “a” $ =\dfrac{x}{4x}=\dfrac{1}{4} $
Similarly, we are going to convert the ratio of odds against winning the race for horse “b” having ratio 4:1.
Unfavorable outcomes are equal to 4x and favorable outcomes are equal to x. From these outcomes, the total outcomes will be 5x.
So, substituting these values in the formula of probability of winning the race by the horse we get,
Probability of winning the race by horse “b” $ =\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}} $
Probability of winning the race by horse “b” $ =\dfrac{x}{5x}=\dfrac{1}{5} $
Similarly, we are going to convert the ratio of odds against winning the race for horse “c” having ratio 5:1.
Unfavorable outcomes are equal to 5x and favorable outcomes are equal to x. From these outcomes, the total outcomes will be 6x.
So, substituting these values in the formula of probability of winning the race by the horse we get,
Probability of winning the race by horse “c” $ =\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}} $
Probability of winning the race by horse “c” $ =\dfrac{x}{6x}=\dfrac{1}{6} $
Now, we are asked to find the probability of winning any one of the horses which is the addition of the probability of winning the race for 3 horses “a, b and c”.
Probability of winning any one of the three horses is equal to:
  $ \dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6} $
Taking L.C.M of the above expression we get,
 $ \begin{align}
  & \dfrac{15+12+10}{60} \\
 & =\dfrac{37}{60} \\
\end{align} $
Hence, the probability of winning any of the three horses is $ \dfrac{37}{60} $ .

Note: You might be thinking that in finding the probability of winning any of three horses why we have just added the winning probabilities of three horses because the probabilities of the winning of three horses is mutually exclusive from each other.
There is a trick to remember the meaning of “odds against an event” that as the statement contains “against” so in the numerator unfavorable outcomes will come which will be divided by favorable outcomes.