Question

The probabilities that Mr. A and Mr. B will dies within a year are $\dfrac{1}{2}$ and $\dfrac{1}{3}$ respectively, then the probability that only one of them will be alive at the end of the year, is
A. $\dfrac{5}{6}$
B. $\dfrac{1}{2}$
C. $\dfrac{2}{3}$
D. None of the above

Answer 1

Hint: Use the concept and formulae for finding the probability of only one of them will be alive at the end of the year. For that we find the probability of A will die and B alive or B will die and A is alive and then use the concept of mutually exclusive events to get to the final answer.

Complete step-by-step solution:
Let us consider that Mr. A will die within a year and denote this event by X. Let us consider that Mr. B will die within a year and denote this event by Y.

Then we apply the probability formulae to find the probability of A will die and B alive or B will die and A is alive.
$
   \Rightarrow P\left[ {\left( {{\text{A will die and B alive}}} \right){\text{ or }}\left( {{\text{B will die and A alive}}} \right)} \right] \\
   = P\left[ {\left( {X \cap Y'} \right) \cup P\left( {Y \cap X'} \right)} \right] \\
$

Now, by the nature of the events they are mutually exclusive events, thus,
$ \Rightarrow P\left( {X \cap Y'} \right) + P\left( {Y \cap X'} \right)$

Now, also we note that the events X and Y are independent of each other.
$ \Rightarrow P\left( X \right).P\left( {Y'} \right) + P\left( Y \right) \cdot P\left( {X'} \right)$ ……(1)

From the given values, and using the fact that $P\left( E \right) + P\left( {{\text{ not }}E} \right) = 1$ , find the value of the unknowns.
\[
  P\left( X \right) = \dfrac{1}{2}; \\
   \Rightarrow P\left( {X'} \right) = 1 - \dfrac{1}{2} \\
   \Rightarrow P\left( {X'} \right) = \dfrac{1}{2} \\
\]
Also,
$
  P\left( Y \right) = \dfrac{1}{3} \\
   \Rightarrow P\left( {Y'} \right) = 1 - \dfrac{1}{3} \\
   \Rightarrow P\left( {Y'} \right) = \dfrac{2}{3} \\
 $

Substitute all these values into (1).
$
   \Rightarrow \dfrac{1}{2}.\dfrac{2}{3} + \dfrac{1}{3} \cdot \dfrac{1}{2} = \dfrac{{2 + 1}}{6} \\
   = \dfrac{3}{6} \\
   = \dfrac{1}{2} \\
$
Thus, the probability that only one of them will be alive at the end of the year is $\dfrac{1}{2}$. Hence, option (B) is the correct option.

Note: Try and find the probability of the event happening and an event not happening. Consider the condition given and asked in the question, and always try to analyze if there is a relation between them. If yes, then based on the given conditions, determine how from the given condition you can reach to the required value. Avoid making any calculation mistakes. Also, make your concept clear on the concepts of mutually exclusive and independent events.