Good question! It uses the concept of memorylessness in statistics and probability.
Let's go slowly and collect facts and analyze. ( IMHO, collecting clear facts from a probability question gives us 50% confidence. )
- Let $x$ be the time that the first person leaves the system.
- Let $Y$ be the total waiting time that the remaining person has (starting at time 0)
- Let $Z$ be the total waiting time that Miss. Granger has (starting at time $x$ ).
Then the probability that Miss. Granger is not the last person to leave the post office is :
$P(Y \geq Z + x | Y \geq x )$
The memoryless property of the exponential says that, if X is exponentially distributed, then,
$P(X \geq x + y | X \geq x) = P(X \geq y)$
Substituting X = Y, y = Z, into the above equation we find that,
$P(Y ≥ Z + x | Y \geq x) = P(Y \geq Z)$
Now, Y and Z are both exponentially distributed random variables with parameter $ \lambda $, so $ P(Y \geq Z) = \frac{1}{2} $. (you can also verify this by symmetry.)
Finally, the answer is $ \frac{1}{2} $
-sudoankit