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

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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) $frac12$



B) $frac14$



C) $frac13$



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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  • 3




    The essential idea is, the probability of anything happening at any exact time is basically 0. This is the reason for the integral logic.
    – Rushabh Mehta
    Aug 17 at 0:17










  • 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)?".
    – CompuChip
    Aug 17 at 7:07






  • 1




    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.
    – Asaf Karagila♦
    Aug 17 at 7:18














up vote
1
down vote

favorite













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




A) $frac12$



B) $frac14$



C) $frac13$



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 ?










share|cite|improve this question



















  • 3




    The essential idea is, the probability of anything happening at any exact time is basically 0. This is the reason for the integral logic.
    – Rushabh Mehta
    Aug 17 at 0:17










  • 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)?".
    – CompuChip
    Aug 17 at 7:07






  • 1




    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.
    – Asaf Karagila♦
    Aug 17 at 7:18












up vote
1
down vote

favorite









up vote
1
down vote

favorite












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




A) $frac12$



B) $frac14$



C) $frac13$



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 ?










share|cite|improve this question
















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




A) $frac12$



B) $frac14$



C) $frac13$



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 ?







probability probability-distributions






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edited Aug 17 at 7:17









Asaf Karagila♦

295k32410738




295k32410738










asked Aug 17 at 0:04









Sheldon

161




161







  • 3




    The essential idea is, the probability of anything happening at any exact time is basically 0. This is the reason for the integral logic.
    – Rushabh Mehta
    Aug 17 at 0:17










  • 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)?".
    – CompuChip
    Aug 17 at 7:07






  • 1




    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.
    – Asaf Karagila♦
    Aug 17 at 7:18












  • 3




    The essential idea is, the probability of anything happening at any exact time is basically 0. This is the reason for the integral logic.
    – Rushabh Mehta
    Aug 17 at 0:17










  • 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)?".
    – CompuChip
    Aug 17 at 7:07






  • 1




    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.
    – Asaf Karagila♦
    Aug 17 at 7:18







3




3




The essential idea is, the probability of anything happening at any exact time is basically 0. This is the reason for the integral logic.
– Rushabh Mehta
Aug 17 at 0:17




The essential idea is, the probability of anything happening at any exact time is basically 0. This is the reason for the integral logic.
– Rushabh Mehta
Aug 17 at 0:17












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)?".
– CompuChip
Aug 17 at 7:07




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)?".
– CompuChip
Aug 17 at 7:07




1




1




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.
– Asaf Karagila♦
Aug 17 at 7:18




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.
– Asaf Karagila♦
Aug 17 at 7:18










3 Answers
3






active

oldest

votes

















up vote
6
down vote













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 $5.0dots01$ 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.






share|cite|improve this answer






















  • In fact, measuring what a "day" really is is a whole different problem
    – vrugtehagel
    Aug 17 at 1:42










  • 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.
    – Robert Israel
    Aug 17 at 1:58










  • 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.
    – Robert Israel
    Aug 17 at 2:04










  • @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. :)
    – Caramiriel
    Aug 17 at 7:20

















up vote
2
down vote













Since as you say the probability is the area under the curve of the PDF, then
$$P(a leq X leq b) = intlimits_a^b f_X(x) dx$$
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
$$P(X = x) = P(a leq X leq a) = intlimits_a^a f_X(x) dx = 0$$






share|cite|improve this answer



























    up vote
    2
    down vote













    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.






    share|cite|improve this answer






















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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      6
      down vote













      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 $5.0dots01$ 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.






      share|cite|improve this answer






















      • In fact, measuring what a "day" really is is a whole different problem
        – vrugtehagel
        Aug 17 at 1:42










      • 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.
        – Robert Israel
        Aug 17 at 1:58










      • 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.
        – Robert Israel
        Aug 17 at 2:04










      • @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. :)
        – Caramiriel
        Aug 17 at 7:20














      up vote
      6
      down vote













      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 $5.0dots01$ 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.






      share|cite|improve this answer






















      • In fact, measuring what a "day" really is is a whole different problem
        – vrugtehagel
        Aug 17 at 1:42










      • 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.
        – Robert Israel
        Aug 17 at 1:58










      • 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.
        – Robert Israel
        Aug 17 at 2:04










      • @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. :)
        – Caramiriel
        Aug 17 at 7:20












      up vote
      6
      down vote










      up vote
      6
      down vote









      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 $5.0dots01$ 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.






      share|cite|improve this 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 $5.0dots01$ 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.







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited Aug 17 at 0:17

























      answered Aug 17 at 0:10









      Robert Israel

      308k22201444




      308k22201444











      • In fact, measuring what a "day" really is is a whole different problem
        – vrugtehagel
        Aug 17 at 1:42










      • 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.
        – Robert Israel
        Aug 17 at 1:58










      • 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.
        – Robert Israel
        Aug 17 at 2:04










      • @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. :)
        – Caramiriel
        Aug 17 at 7:20
















      • In fact, measuring what a "day" really is is a whole different problem
        – vrugtehagel
        Aug 17 at 1:42










      • 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.
        – Robert Israel
        Aug 17 at 1:58










      • 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.
        – Robert Israel
        Aug 17 at 2:04










      • @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. :)
        – Caramiriel
        Aug 17 at 7:20















      In fact, measuring what a "day" really is is a whole different problem
      – vrugtehagel
      Aug 17 at 1:42




      In fact, measuring what a "day" really is is a whole different problem
      – vrugtehagel
      Aug 17 at 1:42












      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.
      – Robert Israel
      Aug 17 at 1:58




      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.
      – Robert Israel
      Aug 17 at 1:58












      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.
      – Robert Israel
      Aug 17 at 2:04




      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.
      – Robert Israel
      Aug 17 at 2:04












      @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. :)
      – Caramiriel
      Aug 17 at 7:20




      @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. :)
      – Caramiriel
      Aug 17 at 7:20










      up vote
      2
      down vote













      Since as you say the probability is the area under the curve of the PDF, then
      $$P(a leq X leq b) = intlimits_a^b f_X(x) dx$$
      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
      $$P(X = x) = P(a leq X leq a) = intlimits_a^a f_X(x) dx = 0$$






      share|cite|improve this answer
























        up vote
        2
        down vote













        Since as you say the probability is the area under the curve of the PDF, then
        $$P(a leq X leq b) = intlimits_a^b f_X(x) dx$$
        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
        $$P(X = x) = P(a leq X leq a) = intlimits_a^a f_X(x) dx = 0$$






        share|cite|improve this answer






















          up vote
          2
          down vote










          up vote
          2
          down vote









          Since as you say the probability is the area under the curve of the PDF, then
          $$P(a leq X leq b) = intlimits_a^b f_X(x) dx$$
          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
          $$P(X = x) = P(a leq X leq a) = intlimits_a^a f_X(x) dx = 0$$






          share|cite|improve this answer












          Since as you say the probability is the area under the curve of the PDF, then
          $$P(a leq X leq b) = intlimits_a^b f_X(x) dx$$
          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
          $$P(X = x) = P(a leq X leq a) = intlimits_a^a f_X(x) dx = 0$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 17 at 0:09









          Ahmad Bazzi

          6,2091624




          6,2091624




















              up vote
              2
              down vote













              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.






              share|cite|improve this answer


























                up vote
                2
                down vote













                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.






                share|cite|improve this answer
























                  up vote
                  2
                  down vote










                  up vote
                  2
                  down vote









                  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.






                  share|cite|improve this answer














                  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.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Aug 17 at 1:39

























                  answered Aug 17 at 0:21









                  Ethan Bolker

                  36.7k54299




                  36.7k54299



























                       

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