Question

The probability of a team winning the math is $\dfrac{P}{{15}}$, the match cannot end in a tie, and the probability of the team losing the match is $\dfrac{2}{5}$. What is the value of $P$.

1. 5
2. 2
3. 1
4. 9

Answer 1

Hint: First we will find the possible outcomes from the given question. Then, according to the question, the sum of probability of winning and probability of losing will be equal to 1. Hence, use the values given in the question to solve for $P$.

Complete step by step solution:
The probability of all the events defined in the sample space add up to 1, that is the probability of an event happening and that event not happening (given it is the only option other than the event) adds to 1.
That is, ${\text{Probability}}\left( a \right) + {\text{Probability}}\left( {{\text{not }}a} \right) = 1$
According to the question, there can be only two possible outcomes, that is the team can either win or lose.
Here the probability of a team winning the match is $\dfrac{P}{{15}}$ and the team not winning the match is $\dfrac{2}{5}$.
Since these are the only two possible outcomes and both the events cannot occur at the same time, thus, the sum of their probabilities must add up to 1.
Therefore we can say that, ${\text{Probability}}\left( {{\text{winning}}} \right) + {\text{Probability}}\left( {{\text{losing}}} \right) = 1$
$\dfrac{P}{{15}} + \dfrac{2}{5} = 1$
On multiplying the equation throughout by 15, we can solve the equation for $P$
$P + 6 = 15 \\\ P = 15 - 6 \\\ P = 9 \\\$
Hence, the value of $P$ is 9.
Thus the correct option is D.

Note: The events in the question cannot occur at the same time, hence, these are known as mutually exclusive events. Since there are only two events possible and are mutually exclusive to each other, therefore, the sum of the probabilities of the events will be equal to 1.