Question

In a certain race, there are three boys A, B, C. The winning probability of A is twice than B and the winning probability of B is twice than C. If \[P\left( A \right) + P\left( B \right) + P\left( C \right){\rm{ = 1}}\], then find the probability of a win for each boy.

Answer 1

Hint:
Here, we have to use the concept of probability to find out the probability of a win for each boy. Firstly we will assume the probability of a win for boy C and use the given condition to get all the probabilities in terms of C. Then by putting all the values of the probability in the given equation of the question, we will get the value of the probability of a win for each boy.

Complete step by step solution:
Let us assume the probability of winning for boy C be \[{\rm{P}}\left( {\rm{C}} \right) = x\].
Now we have to note all the given information from the question and make it in the form of the assumed probability of winning for boy C, we get
According to the question it is given that the winning probability of B is twice than C. So we can write,
\[\begin{array}{l}P\left( B \right) = 2 \times P\left( C \right)\\ \Rightarrow P\left( B \right) = 2x\end{array}\]
Also, it is given that the winning probability of A is twice than B.
So,
 \[\begin{array}{l}P\left( A \right) = 2 \times P\left( B \right)\\ \Rightarrow P\left( A \right) = 2 \times 2x = 4x\end{array}\]
Now we will substitute the values of all the probability in the equation, \[P\left( A \right) + P\left( B \right) + P\left( C \right) = 1\]. Therefore, we get
\[\begin{array}{l}P\left( A \right) + P\left( B \right) + P\left( C \right) = 1\\ \Rightarrow 4x + 2x + x = 1\end{array}\]
Adding the like terms, we get
\[ \Rightarrow 7x = 1\]
Dividing both side of the equation by 7, we get
\[ \Rightarrow x = \dfrac{1}{7}\]
Therefore, \[P\left( C \right) = \dfrac{1}{7}\]
Now by using the value of \[x\] we will get the value of \[P\left( A \right)\], \[P\left( B \right)\].
Therefore, we get
\[P\left( A \right) = 4x = 4 \times \dfrac{1}{7} = \dfrac{4}{7}\]
\[P\left( B \right) = 2x = 2 \times \dfrac{1}{7} = \dfrac{2}{7}\]

Hence, \[P\left( A \right) = \dfrac{4}{7}\], \[P\left( B \right) = \dfrac{2}{7}\]and \[P\left( C \right) = \dfrac{1}{7}\].

Note:
Probability is the branch of mathematics which gives the possibility of the event occurrence and is equal to the ratio of the number of favorable outcomes to the total number of outcomes. Probability generally lies between the values of 0 to 1. Similarly in the above question, we got the probability in the range of 0 and 1. Probability can never be above the value of 1 or less than 0. So if we got the value of probability greater than 1 then we need to check out the calculation to remove the error for calculating the probability.