Question

A, B, C are three horses in a race. The probability of A to win the race is twice that of B and probability of B is twice that of C. What are the probability of A, B and C to win the race?

Answer 1

Hint: With the given question we will initially find the probability of C then with the help of it we will find the probability of other horses too.

Formula used: Let us consider, the total number of outcomes of an event space is \[n\]. And, the number of outcomes for a particular event is \[m.\]
So, \[{\text{The probability of a certain event P is = }}\dfrac{{{\text{the number of outcomes for a particular event\;}}}}{{{\text{the total number of outcomes of an event space}}}}\]
The total probability of different probability will be 1.

Complete step-by-step answer:
It is given that: A, B, C are three horses in a race. The probability of A to win the race is twice that of B and probability of B is twice that of C.
Let us consider, the probability of winning of horse C is \[x\]
${\text{x = }}\dfrac{{{\text{the number of outcomes winning of horse C}}}}{{{\text{the total number of outcomes of an event space}}}}$
Then, the probability of winning of horse B is \[2x\] let it be condition (i)
\[{\text{the probability of winning of horse B = 2(the probability of winning of horse C)}}\]
And, the probability of winning of horse A is\[4x\] let it be condition (ii)
\[{\text{the probability of winning of horse A = 4(the probability of winning of horse C)}}\]
We know that,
The total probability of different probability will be 1.
So, from the hints given in the question, we have
\[x + 2x + 4x = 1\]
Adding every term in the left hand side we get,
\[7x = 1\]
Now let us simplify again to get value of x, we get,
\[x = \dfrac{1}{7}\]
Therefore,
The probability of winning on horse C is\[\dfrac{1}{7}\].
Now by substituting the value of x in condition (i) we get,
The probability of winning of horse B is \[\dfrac{2}{7}\]
Now by substituting the value of x in condition (ii) we get,
The probability of winning horse A is \[\dfrac{4}{7}\].
Hence we have found the probability of the three horses.

Note:Here, A, B, C are mutually exclusive events. That means there is no other possibility of two or all of them winning together.
Again if A, B, C are exhaustive events then that means there are no horses running the race.