Question

The probability that A can win a race is $\dfrac{3}{8}$ and probability that B can win it is $\dfrac{1}{8}$. If both run in a race, find the probability that one of them will win the race assume that both cannot win together.
A) $0$
B) $\dfrac{{15}}{{24}}$
C) $\dfrac{{13}}{{24}}$
D) $\dfrac{1}{4}$

Answer 1

Hint:
Use the basic rule of the probability. In probability we use three basic rules that is addition, multiplication, and complement rules. The addition rule is used to find the probability of event A or event B happening such as ${\rm{P}}\left( {{\rm{A}} \cup {\rm{B}}} \right) = {\rm{P}}\left( {\rm{A}} \right) + {\rm{P}}\left( {\rm{B}} \right) - {\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right)$.

Complete step by step solution:
We know from the question that the probability that A can win a race is ${\rm{P}}\left( {\rm{A}} \right) = \dfrac{3}{8}$ and the probability that B can win it is ${\rm{P}}\left( {\rm{B}} \right) = \dfrac{1}{8}$.
Assume that the both cannot win the race together as ${\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right) = 0$.
Now, from the below relation we can calculate the probability that one of them will win the race.
${\rm{P}}\left( {{\rm{A}} \cup {\rm{B}}} \right) = {\rm{P}}\left( {\rm{A}} \right) + {\rm{P}}\left( {\rm{B}} \right) - {\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right)$
Substitute the values ${\rm{P}}\left( {\rm{A}} \right)$ as $\dfrac{3}{8}$, ${\rm{P}}\left( {\rm{B}} \right)$ as $\dfrac{1}{8}$ and ${\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right)$ as 0 in the above equation.
$
{\rm{P}}\left( {{\rm{A}} \cup {\rm{B}}} \right) = {\rm{P}}\left( {\rm{A}} \right) + {\rm{P}}\left( {\rm{B}} \right) - {\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right)\\
 = \dfrac{3}{8} + \dfrac{1}{6} - 0\\
 = \dfrac{{18 + 8}}{{48}}\\
 = \dfrac{{13}}{{24}}
$

Hence, from the above result we can say the probability that one of them will win the race is $\dfrac{{13}}{{24}}$ and the option (C) is correct.

Additional Information:
The formula ${\rm{P}}\left( {{\rm{A}} \cup {\rm{B}}} \right) = {\rm{P}}\left( {\rm{A}} \right) + {\rm{P}}\left( {\rm{B}} \right) - {\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right)$ is very popular when the questions of venn diagram with probability is asked in exams. So make sure that the concept of this type of problem is clear. Venn diagram problems look tricky but are very simple with its help.

Note:
Make sure that the assumption of both not A and B not winning the race is necessary to solve the question. Here we use the basic probability rules, if there are two events like A and B then ${\rm{P}}\left( {{\rm{A}} \cup {\rm{B}}} \right) = {\rm{P}}\left( {\rm{A}} \right) + {\rm{P}}\left( {\rm{B}} \right) - {\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right)$. Just a method to remember this formula is, it’s similar to the formula of number of elements in the union of two sets.