Question

Three athletes A, B and C participate in a race competition. The probability of winning for A and B is twice of winning for C. Then the probability that the race is won by A or B, is
$1)\dfrac{2}{3}$
$2)\dfrac{3}{4}$
$3)\dfrac{4}{5}$
$4)\dfrac{1}{6}$

Answer 1

Hint:To solve this question we need to have the knowledge of probability. The first step will be to consider the probability of C winning the race be “x” so the probability of A and B will be equal to 2x the next step will be to take the sum of the probability of all the three winning the race to be $1$. With the help of the above condition we will find the probability of A and B winning the race.

Complete step-by-step solution:
The question ask us to find the probability of the race run by A or B if three athletes A, B and C participate in a race competition and the condition here are given is that the probability of winning the race by A and B is twice that of winning by C. The condition when written mathematically looks like:
$\Rightarrow P(A)=P(B)=2P(C)$
In the above expression $P(A)$ represents the probability of A winning the race, similar explanation is for both $P(B)$ and $P(C)$. The next step is to add the probability of the three athletes to sum up to $1$. On doing this mathematically we get:
$\Rightarrow P(A)+P(B)+P(C)=1$
We will write the probability of A and B in terms of the probability of C. On writing this mathematically we get:
$\Rightarrow 2P(C)+2P(C)+P(C)=1$
Adding up all the common terms we get:
$\Rightarrow 5P(C)=1$
On calculating it further we get:
$\Rightarrow P(C)=\dfrac{1}{5}$
So the probability of A and B will be:
$\Rightarrow P(A)=P(B)=2P(C)=\dfrac{2}{5}$
The probability of winning the race by A or B will be:
$\Rightarrow P(A)orP(B)$
$\Rightarrow P(A)+P(B)$
$\Rightarrow \dfrac{2}{5}+\dfrac{2}{5}$
$\Rightarrow \dfrac{4}{5}$
$\therefore $ Then the probability that the race is won by A or B, is $3)\dfrac{4}{5}$ .

Note:Always remember that the probability of the event is always between $0$ and $1$. When the probability of the event is $1$, it means the event is certain as it could be seen in the solution above as winning of any of the athletes is certain the sum of the probability was considered $1$.