Question

If the probability of horse A winning a race is \[\dfrac{1}{4}\] and the probability of a horse B winning a race is \[\dfrac{1}{5}\], then the probability that either of them will win the race is
1. \[\dfrac{1}{20}\]
2. \[\dfrac{9}{20}\]
3. \[\dfrac{11}{20}\]
4. \[\dfrac{19}{20}\]

Answer 1

Hint: In this problem, we have to find the probability of either two horses winning the race where we are given the probabilities of each horse winning. We can first write the formula of the probability of either winning. We can then find whether they are mutually exclusive and substitute the given probabilities in the formula and we can find the answer.

Complete step by step answer:
Here we are given that the probability of horse A winning a race is \[\dfrac{1}{4}\] and the probability of a horse B winning a race is \[\dfrac{1}{5}\].
\[\Rightarrow P\left( A \right)=\dfrac{1}{4},P\left( B \right)=\dfrac{1}{5}\] …… (1)
We know that,
The probability that either of them wins the race = \[P\left( A\cup B \right)\].
We also know that,
\[P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)\] ……. (2)
We know that only one of the two horses can win the race, so event A and B are mutually exclusive events.
\[\Rightarrow P\left( A\cap B \right)=0\] …….. (3)
We can now substitute (1) and (3) in (2), we get
\[\begin{align}
  & \Rightarrow P\left( A\cup B \right)=\dfrac{1}{4}+\dfrac{1}{5}-0 \\
 & \Rightarrow P\left( A\cup B \right)=\dfrac{5+4}{20}=\dfrac{9}{20} \\
\end{align}\]

So, the correct answer is “Option 2”.

Note: We should always remember that if the given two events are mutually exclusive events then \[P\left( A\cap B \right)=0\], where the two events do not occur at the same time and only one event can occur. We should also remember that \[P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)\].