A fly has a lifespan of $4$–$6$ days. What is the probability that the fly will die at exactly $5$ days?

Question

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 ?

Answer 1

Since as you say the probability is the area under the curve of the PDF, then $$P(a \leq X \leq b) = \int\limits_a^b f_X(x) \ dx$$ where $f_X(x)$ is the PDF of $X$, which is the random variable describing the probability that the fly will die at time $X = x$. Now since you are interested in $P(X = a)$, then $$P(X = x) = P(a \leq X \leq a) = \int\limits_a^a f_X(x) \ dx = 0$$

Answer 2

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.

Answer 3

The answers that use integrals to explain the correct $0$ answer are correct. I offer a more intuitive reason.

The key word in the question is "exactly". Since there are infinitely many possible exact times the fly could die, if the times were all equally probable there is no way that the probability of an exact time can be greater than $0$ since the sum of the probabilities must be $1$, which is finite.

The question does not say all times are equally probable, but it does say that the probability distribution is continuous. Then too no exact time can have a nonzero probability, because continuity would imply that infinitely many nearby times had at least half that nonzero probability so the sum would again be infinite.

The "come to think of it" at the end of the answer points to the artificiality of the question. Using a continuous distribution to model a genuine physical (or in this case biological) problem forces you to confront questions about precision of measurement.