A fly has a lifespan of $4$–$6$ days. What is the probability that the fly will die at exactly $5$ days?
A) $\frac{1}{2}$
B) $\frac{1}{4}$
C) $\frac{1}{3}$
D) $0$
The solution given is
"Here since the probabilities are continuous, the probabilities form a mass function. The probability of a certain event is calculated by finding the area under the curve for the given conditions. Here since we’re trying to calculate the probability of the fly dying at exactly 5 days – the area under the curve would be 0. Also to come to think of it, the probability if dying at exactly 5 days is impossible for us to even figure out since we cannot measure with infinite precision if it was exactly 5 days."
But I do not seem to understand the solution, can anyone help here ?
Best Answer
If the random variable $X$ has a continuous probability distribution with probability density function $f$, the probability of $X$ being in the interval $[a,b]$ is $$ \mathbb P(a \le X \le b) = \int_a^b f(x)\; dx$$ i.e. the area under the curve $y = f(x)$ for $x$ from $a$ to $b$. But if $a = b$ that area and that integral, are $0$.
Note that "exactly" in mathematics is very special. If the fly's lifetime is $5.0\dots01$ days (with as many zeros as you wish), it does not count as "exactly" $5$ days. But we can't actually measure the fly's lifetime with perfect precision, so we could never actually say that the fly's lifetime was exactly $5$ days.