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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 ?

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    $\begingroup$ The essential idea is, the probability of anything happening at any exact time is basically 0. This is the reason for the integral logic. $\endgroup$ – Don Thousand Aug 17 '18 at 0:17
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    $\begingroup$ Does anyone else find the phrasing confusing? "The fly will die at midnight on the fifth day" clearly refers to one time instant and then the reasoning given is correct, but initially I read "exactly five days" as meaning "not on the fourth or sixth day" and figured "it could be any of A) - C) ... perhaps they forgot to state the the probability distribution is uniform and it should be C)?". $\endgroup$ – CompuChip Aug 17 '18 at 7:07
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    $\begingroup$ Please make your question's body self-contained. If you need to write the same thing in the title and in the body, that's a better option. $\endgroup$ – Asaf Karagila Aug 17 '18 at 7:18
  • $\begingroup$ This is kind of a trick question. I initially read it as having a normal distribution where 95% (2 SD) of the population dies between 4 and 6 days. What percentage dies on day 5 between 00:00:00 and 23:59:59.999+. When they say "exactly," all that goes out the window and zero is the only answer. The probability that any will die on day 5 between 12:00:00 and 12:00:00.000001 is low (I get 6E-12). "Exactly" means effectively zero time so the probability is zero. $\endgroup$ – user1683793 Dec 15 '18 at 23:56
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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.

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  • $\begingroup$ In fact, measuring what a "day" really is is a whole different problem $\endgroup$ – vrugtehagel Aug 17 '18 at 1:42
  • $\begingroup$ A day is $86400$ seconds, where the official (SI) definition of a second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. $\endgroup$ – Robert Israel Aug 17 '18 at 1:58
  • $\begingroup$ More problematical is pinpointing the exact beginning and end of the fly's life, since birth (or in this case hatching) and death are gradual processes. $\endgroup$ – Robert Israel Aug 17 '18 at 2:04
  • $\begingroup$ @RobertIsrael Since "A day is 86400 seconds", you'd not be able to answer "on what day is daylight saving time changed?" because that day doesn't have exactly 86400 seconds. But yeah, what vrugtehagel said seems about right. :) $\endgroup$ – Caramiriel Aug 17 '18 at 7:20
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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 $$P(X = x) = P(a \leq X \leq a) = \int\limits_a^a f_X(x) \ dx = 0$$

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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.

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