Question

In a horse race the odds in favour of three horses are 1:2, 1:3, and 1:4. What is the probability that one of the horses will win the race?
A. \[\dfrac{7}{{60}}\]
B. \[\dfrac{{47}}{{60}}\]
C. \[\dfrac{1}{4}\]
D. \[\dfrac{3}{4}\]

Answer 1

Hint Odds in favour means the ratio of the number of outcomes of an event to the number of outcomes that the event is not happening. By using the odds in favour, we calculate the probability of winning each horse. Then add up all probability to get the result.

Formula used
\[{\rm{Probability = }}\dfrac{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}}}{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{total}}\,{\rm{outcomes}}}}\]
Probability of independent events: \[P\left( {A \cup B \cup C} \right) = P\left( A \right) + P\left( B \right) + P\left( C \right)\]
\[{\rm{Probability = }}\dfrac{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}}}{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{total}}\,{\rm{outcomes}}}}\]

Complete step by step solution:
Given that, in a horse race, the odds in favour of three horses are 1:2, 1:3, and 1:4.
Odds in favour means the ratio of the number of outcomes of an event to the number of outcomes that the event is not happening.
The number of favorable outcomes of the first horse is 1.
The total number of outcomes for the first horse is \[1 + 2 = 3\]
Applying the formula \[{\rm{Probability = }}\dfrac{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}}}{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{total}}\,{\rm{outcomes}}}}\] to find the probability of wining of the first horse
The probability of winning the first horse is \[\dfrac{1}{3}\].
The number of favorable outcomes of the second horse is 1.
The total number of outcomes for the second horse is \[1 + 3 = 4\]
Applying the formula \[{\rm{Probability = }}\dfrac{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}}}{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{total}}\,{\rm{outcomes}}}}\] to find the probability of wining the second horse
The probability of winning the second horse is \[\dfrac{1}{4}\].
The number of favorable outcomes of the second horse is 1.
The total number of outcomes for the second horse is \[1 + 4 = 5\]
Applying the formula \[{\rm{Probability = }}\dfrac{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}}}{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{total}}\,{\rm{outcomes}}}}\] to find the probability of wining the second horse
The probability of winning the second horse is \[\dfrac{1}{5}\].
 Now apply the formula \[P\left( {A \cup B \cup C} \right) = P\left( A \right) + P\left( B \right) + P\left( C \right)\] to calculate the probability that one of the horses will win the race.
The probability that one of the horses will win the race is \[\dfrac{1}{3} + \dfrac{1}{4} + \dfrac{1}{5}\]
 \[ = \dfrac{{20 + 15 + 12}}{{60}}\]
\[ = \dfrac{{47}}{{60}}\]
Hence option B is the correct option.

Note: Many students make a common mistake to calculate the probability that one of the horses will win the race. They add odd in favour to calculate the probability. For this reason, they get the wrong answer.