Question

The probability that a man who is 85 years old will die before attaining the age of 90 is $\dfrac{1}{3}$. The four persons ${A_1}$, ${A_2}$, ${A_3}$, and ${A_4}$are 85 years old. The probability that ${A_1}$ will die before attaining the age of 90 and will be the first to die is:
A. $\dfrac{{65}}{{81}}$
B. $\dfrac{{13}}{{81}}$
C. $\dfrac{{65}}{{324}}$
D. $\dfrac{{13}}{{108}}$

Answer 1

Hint: Here, we will use the formula of finding the probability of any event happening by dividing the number of favorable outcomes of that event divided by the total number of events, that is;
$P = \dfrac{{{\text{Number of favorable outcomes}}}}{{{\text{Total number of outcomes}}}}$
Apply this formula, and then use the given conditions to find the required value.

Complete step-by-step answer
Given that the probability that a man who is 85 years old will die before attaining the age $B$ of 90 is $\dfrac{1}{3}$, that is, ${\text{P}}\left( B \right) = \dfrac{1}{3}$.

We know that there are four persons ${A_1}$, ${A_2}$, ${A_3}$, and ${A_4}$, who are 85 years old.

Since there are 4 persons and they have the same probability to die, the probability that ${A_1}$ dies first is $\dfrac{1}{4}$.

We know that the probability of favorable choice is ${\text{P}}\left( {\text{E}} \right)$ and probability of unfavorable choice is ${\text{P}}\left( {\overline {\text{E}} } \right)$, then ${\text{P}}\left( {\text{E}} \right) + {\text{P}}\left( {\overline {\text{E}} } \right) = 1$.

We will now find the probability that the person dies after the age of 90, that is, ${\text{P}}\left( {\overline B } \right)$ using the above property of probability.

${\text{P}}\left( B \right) + {\text{P}}\left( {\overline B } \right) = 1$

Substituting the value of ${\text{P}}\left( B \right)$ in the above equation, we get

$$\Rightarrow \dfrac{1}{3} + {\text{P}}\left( {\overline B } \right) = 1 \\\ \Rightarrow {\text{P}}\left( {\overline B } \right) = 1 - \dfrac{1}{3} \\\ \Rightarrow {\text{P}}\left( {\overline B } \right) = \dfrac{{3 - 1}}{3} \\\ \Rightarrow {\text{P}}\left( {\overline B } \right) = \dfrac{2}{3} \\\$$

Now we will find the probability that all the four persons will die after the age of 90.

${\left( {\dfrac{2}{3}} \right)^4} = \dfrac{{16}}{{81}}$

Since the two events are independent, we will find the probability that ${A_1}$ dies first and all the four persons die after the age of 90.

$\dfrac{{16}}{{81}} \cdot \dfrac{1}{4} = \dfrac{4}{{81}}$

But if ${A_1}$ dies first, then either he dies first and is above or he dies first and is younger than 90, and these two are mutually exclusive.

We will now use the probability rule for mutually exclusive events, that is, $P\left( X \right) = P\left( {X \cap Y} \right) + P\left( {X \cap {Y^C}} \right)$.

Here, we have $P\left( X \right) = \dfrac{1}{4}$ and $P\left( {X \cap Y} \right) = \dfrac{4}{{81}}$.

We will now find the probability that dies first and before the age of 90 using these values and above rule.

$$P\left( {X \cap {Y^C}} \right) = \dfrac{1}{4} - \dfrac{4}{{81}} \\\ = \dfrac{{69 - 4}}{{324}} \\\ = \dfrac{{65}}{{324}} \\\$$

Thus, the probability that ${A_1}$ will die before attaining the age of 90 and will be the first to die is $\dfrac{{65}}{{324}}$.

Hence, the option C is correct.

Note: In solving these types of questions, you should be familiar with the formula to find the probability of any event happening. Students should know if the probability of favorable choice is ${\text{P}}\left( {\text{E}} \right)$ and probability of unfavorable choice is ${\text{P}}\left( {\overline {\text{E}} } \right)$, then ${\text{P}}\left( {\text{E}} \right) + {\text{P}}\left( {\overline {\text{E}} } \right) = 1$ and the basic property of probability $P\left( X \right) = P\left( {X \cap Y} \right) + P\left( {X \cap {Y^C}} \right)$ for mutually exclusive events. Then use the given conditions and values given in the question, to find the required values. The mutually exclusive events are those events that cannot happen simultaneously. Also, we are supposed to write the values properly to avoid any miscalculation.