Question

The probability that Krishna will be alive 10 years hence is $\dfrac{7}{{15}}$ and that Hair will be alive is $\dfrac{7}{{10}}$. The probability both Krishna and Hari will be dead 10 years hence is
A.$\dfrac{{21}}{{150}}$
B.$\dfrac{{24}}{{150}}$
C.$\dfrac{{49}}{{150}}$
D.$\dfrac{{56}}{{150}}$

Answer 1

Hint: We are given the probability $P\left( K \right) = \dfrac{7}{{15}}$ where K is the event that Krishna will be alive and $P\left( H \right) = \dfrac{7}{{10}}$ where H is the event that Hari will be alive, then we will calculate the probability of the events ${K^C}{\text{ and }}{H^C}$ that Krishna and Hari will be dead in next 10 years individually. Since the events ${K^C}{\text{ and }}{H^C}$ are independent of each other, we will use the formula: $P\left( {{A^C} \cap {B^C}} \right) = P\left( {{A^C}} \right) \cdot P\left( {{B^C}} \right)$ to calculate the probability that both Krishna and Hari will be dead 10 years hence.

Complete step-by-step answer:
We are given that the probability of Krishna being alive 10 years hence is $\dfrac{7}{{15}}$. Let the event that Krishna will be alive denoted by $P\left( K \right)$ and then we can write $P\left( K \right) = \dfrac{7}{{15}}$.
Let the event that Hari will be alive be denoted by $P\left( H \right)$ and we are given that the probability that Hari will be alive 10 years hence is $\dfrac{7}{{10}}$, therefore, we can write $P\left( H \right) = \dfrac{7}{{10}}$.
Now, we need to calculate the probability that both Krishna and Hari will be dead 10 years hence.
Let the event that Krishna will be dead 10 years hence be denoted by ${K^C}$ . The probability that Krishna will be dead 10 years will be represented by $P\left( {{K^C}} \right)$.
Similarly, the event that Hari will be dead after 10 years will be denoted by ${H^C}$ . Then, the probability that Hari will be dead 10 years hence will be $P\left( {{H^C}} \right)$.
We can calculate the value of ${K^C}{\text{ and }}{H^C}$ can be calculated by the formula: $P\left( {{A^C}} \right) = 1 - P\left( A \right)$ where A is the event being done.
Therefore, we can say that
$
   \Rightarrow P\left( {{K^C}} \right) = 1 - P\left( K \right) \\
   \Rightarrow P\left( {{K^C}} \right) = 1 - \dfrac{7}{{15}} = \dfrac{8}{{15}} \\
 $
Therefore, the probability that Krishna will be dead 10 years hence is $\dfrac{8}{{15}}$.
Similarly, we can write that
$
   \Rightarrow P\left( {{H^C}} \right) = 1 - P\left( H \right) \\
   \Rightarrow P\left( {{H^C}} \right) = 1 - \dfrac{7}{{10}} = \dfrac{3}{{10}} \\
 $
Therefore, the probability that Hari will be dead 10 years hence is $\dfrac{3}{{10}}$.
Now, we are required to find the probability that both of them will be dead 10 years hence.
Since the events ${K^C}{\text{ and }}{H^C}$ are independent of each other, we will use the formula: $P\left( {{A^C} \cap {B^C}} \right) = P\left( {{A^C}} \right) \cdot P\left( {{B^C}} \right)$ and here ${A^C} = {K^C}{\text{ and }}{B^C} = {H^C}$
$ \Rightarrow P\left( {{K^C} \cap {H^C}} \right) = P\left( {{K^C}} \right) \cdot P\left( {{H^C}} \right)$
$
   \Rightarrow P\left( {{K^C} \cap {H^C}} \right) = \dfrac{8}{{15}} \cdot \dfrac{3}{{10}} \\
   \Rightarrow P\left( {{K^C} \cap {H^C}} \right) = \dfrac{{24}}{{150}} \\
 $
Hence, the probability that both Krishna and Hari will be dead 10 years is $\dfrac{{24}}{{150}}$.
Therefore, option (B) is correct.

Note: In this question, you may get confused in the step where we have calculated the complement of the events since we have used the derived equation from the statement that probability of happening and the probability of not happening of any event sums equal to 1 i.e. $P\left( A \right) + P\left( {{A^C}} \right) = 1$. Here, we have used the formula $P\left( {{A^C} \cap {B^C}} \right) = P\left( {{A^C}} \right) \cdot P\left( {{B^C}} \right)$ because the probability of both being dead after 10 years was asked. If the probability of either of them or just one of them being dead was asked, then we shouldn’t have used this formula.