Question

There are two people who are going to take part in a race. The probability that the first one will win is $\dfrac{2}{7}$ and that of other winning is $\dfrac{3}{5}$. What is the probability that one of them will win the race?

Answer 1

Hint: In this question we are going to use the formula $P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)$ and then we will put the given values in it but remember that in this question event A and event B are mutually exclusive events (disjoint events only one event happens at a time) which means that $P\left( A\cap B \right)=0$. This formula is from the set theory and is also derived from the Venn diagrams and now the formula transforms into $P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)$ and we have to find the value of $P\left( A\cup B \right)$. For our question, we have event A as first person winning the race and event B as second person winning the race. We will then apply the given probabilities in the formula.
Complete step by step solution:
Now in this question we are given the probabilities of occurrence of event A and occurrence of event B, and we have to find the probability that one of them will win the race.
First of all we need to remember the meaning of symbols,
At first we will see a union symbol which is $A\cup B$, now it means 'Addition' it also means 'or' either of two events is considered in it.
Now we will see the intersection symbol which is $A\cap B$, now it means 'Multiplication' it also means 'and' both the events are considered together at once.
In the question it is asked from us to find the probability of that one of them will win which means finding the probability of $A\cup B$ and as the events are mutually exclusive which means that if one event occurs then the second event will not occur or they are disjoint that is no element is common between their sets of sample space which is $A\cap B=0$. When two events are mutually exclusive simply remember they cannot occur together at once and hence mathematically, $P\left( A\cap B \right)=0$.
Now applying the formula of rule of addition,
$P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)$, this formula can be obtained with the help of set theory and Venn diagrams.
Now as $P\left( A\cap B \right)=0$ because both the events are mutually exclusive then the formula becomes,
$P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)$
Now let us consider that event A is winning the race by person 1 and event B is winning the race by person 2.
Now, probability of person 1 winning the race is $P\left( A \right)=\dfrac{2}{7}$ and probability of person 2 winning the race is $P\left( B \right)=\dfrac{3}{5}$.
Putting the known values in the above transformed formula, we get,
$P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)=\dfrac{2}{7}+\dfrac{3}{5}$
$P\left( A\cup B \right)=\dfrac{10+21}{35}$
$P\left( A\cup B \right)=\dfrac{31}{35}$
Hence the answer is $\dfrac{31}{35}$.

Note: You need to remember the formulas from set theory or just understand the formation of the formula relations by using the Venn diagrams. Remember the meanings of the symbols mentioned in the solution as well as for the other symbols also like negation of an event rule of addition for three events etc.. Memorize all the definitions of types of events like mutually exclusive events, exhaustive events, compound events and many more. In these types of questions there is no need for construction of sample space and you won't be able to construct it too so do not waste your time on thinking about it, but remember sample space is important wherever there is possibility to construct it go for it.