Elementary probability theory and its set-theoretic aspects
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Since the publication of Kolmogorov's Foundations of the theory of probability in 1933, all the concepts of the theory have been redefined in a set-theoretic framework (events, probability as a function on the set of subsets of a universe $Omega$, ...). In particular, the logic of events has become the algebra of subsets of $Omega$.
However, in the elementary applications of the theory (like the ones taught in schools), are these set-theoretic aspects necessary ?
Do you see an exercise where the set-theoretic machinery is necessary, or at least very useful, to get to the solution ?
probability probability-theory
asked Nov 21 at 11:05
Sephi
1056
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2
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Since the publication of Kolmogorov's Foundations of the theory of probability in 1933, all the concepts of the theory have been redefined in a set-theoretic framework (events, probability as a function on the set of subsets of a universe $Omega$, ...). In particular, the logic of events has become the algebra of subsets of $Omega$.
However, in the elementary applications of the theory (like the ones taught in schools), are these set-theoretic aspects necessary ?
Do you see an exercise where the set-theoretic machinery is necessary, or at least very useful, to get to the solution ?
probability probability-theory
asked Nov 21 at 11:05
Sephi
1056
A light hearted comment: doing probability theory without measure theory is like having a building without a foundation!
– Kavi Rama Murthy
Nov 21 at 12:13
Already the formula of total probability is much clearer if you draw $Omega$ and your partition, and in fact students are used to consider set-theoretic facts to gather intuitions on a problem from a young age.
– Gâteau-Gallois
Nov 21 at 17:08
"Taught in schools" ...which schools? :) Venn diagrams and partitions are surely "very useful" in any elementary problems (up to highschool level), but the more advanced stuff I'd think arent necessary as long as the sample space is finite. Once you run into infinite / continuous space you might need a good foundation, or else you end up wasting time arguing about "paradoxes" (conditioning on an event of prob 0, en.wikipedia.org/wiki/Bertrand_paradox_(probability), en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_paradox etc)
– antkam
Nov 21 at 20:28
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Since the publication of Kolmogorov's Foundations of the theory of probability in 1933, all the concepts of the theory have been redefined in a set-theoretic framework (events, probability as a function on the set of subsets of a universe $Omega$, ...). In particular, the logic of events has become the algebra of subsets of $Omega$.
However, in the elementary applications of the theory (like the ones taught in schools), are these set-theoretic aspects necessary ?
Do you see an exercise where the set-theoretic machinery is necessary, or at least very useful, to get to the solution ?
probability probability-theory
asked Nov 21 at 11:05
Sephi
1056
Since the publication of Kolmogorov's Foundations of the theory of probability in 1933, all the concepts of the theory have been redefined in a set-theoretic framework (events, probability as a function on the set of subsets of a universe $Omega$, ...). In particular, the logic of events has become the algebra of subsets of $Omega$.
However, in the elementary applications of the theory (like the ones taught in schools), are these set-theoretic aspects necessary ?
Do you see an exercise where the set-theoretic machinery is necessary, or at least very useful, to get to the solution ?
probability probability-theory
probability probability-theory
asked Nov 21 at 11:05
Sephi
1056
asked Nov 21 at 11:05
Sephi
1056
asked Nov 21 at 11:05
Sephi
1056
asked Nov 21 at 11:05
Sephi
1056
asked Nov 21 at 11:05
Sephi
1056
1056
A light hearted comment: doing probability theory without measure theory is like having a building without a foundation!
– Kavi Rama Murthy
Nov 21 at 12:13
Already the formula of total probability is much clearer if you draw $Omega$ and your partition, and in fact students are used to consider set-theoretic facts to gather intuitions on a problem from a young age.
– Gâteau-Gallois
Nov 21 at 17:08
"Taught in schools" ...which schools? :) Venn diagrams and partitions are surely "very useful" in any elementary problems (up to highschool level), but the more advanced stuff I'd think arent necessary as long as the sample space is finite. Once you run into infinite / continuous space you might need a good foundation, or else you end up wasting time arguing about "paradoxes" (conditioning on an event of prob 0, en.wikipedia.org/wiki/Bertrand_paradox_(probability), en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_paradox etc)
– antkam
Nov 21 at 20:28
add a comment |
A light hearted comment: doing probability theory without measure theory is like having a building without a foundation!
– Kavi Rama Murthy
Nov 21 at 12:13
Already the formula of total probability is much clearer if you draw $Omega$ and your partition, and in fact students are used to consider set-theoretic facts to gather intuitions on a problem from a young age.
– Gâteau-Gallois
Nov 21 at 17:08
"Taught in schools" ...which schools? :) Venn diagrams and partitions are surely "very useful" in any elementary problems (up to highschool level), but the more advanced stuff I'd think arent necessary as long as the sample space is finite. Once you run into infinite / continuous space you might need a good foundation, or else you end up wasting time arguing about "paradoxes" (conditioning on an event of prob 0, en.wikipedia.org/wiki/Bertrand_paradox_(probability), en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_paradox etc)
– antkam
Nov 21 at 20:28
A light hearted comment: doing probability theory without measure theory is like having a building without a foundation!
– Kavi Rama Murthy
Nov 21 at 12:13
A light hearted comment: doing probability theory without measure theory is like having a building without a foundation!
– Kavi Rama Murthy
Nov 21 at 12:13
Already the formula of total probability is much clearer if you draw $Omega$ and your partition, and in fact students are used to consider set-theoretic facts to gather intuitions on a problem from a young age.
– Gâteau-Gallois
Nov 21 at 17:08
Already the formula of total probability is much clearer if you draw $Omega$ and your partition, and in fact students are used to consider set-theoretic facts to gather intuitions on a problem from a young age.
– Gâteau-Gallois
Nov 21 at 17:08
"Taught in schools" ...which schools? :) Venn diagrams and partitions are surely "very useful" in any elementary problems (up to highschool level), but the more advanced stuff I'd think arent necessary as long as the sample space is finite. Once you run into infinite / continuous space you might need a good foundation, or else you end up wasting time arguing about "paradoxes" (conditioning on an event of prob 0, en.wikipedia.org/wiki/Bertrand_paradox_(probability), en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_paradox etc)
– antkam
Nov 21 at 20:28
"Taught in schools" ...which schools? :) Venn diagrams and partitions are surely "very useful" in any elementary problems (up to highschool level), but the more advanced stuff I'd think arent necessary as long as the sample space is finite. Once you run into infinite / continuous space you might need a good foundation, or else you end up wasting time arguing about "paradoxes" (conditioning on an event of prob 0, en.wikipedia.org/wiki/Bertrand_paradox_(probability), en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_paradox etc)
– antkam
Nov 21 at 20:28
add a comment |
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A light hearted comment: doing probability theory without measure theory is like having a building without a foundation!
– Kavi Rama Murthy
Nov 21 at 12:13
Already the formula of total probability is much clearer if you draw $Omega$ and your partition, and in fact students are used to consider set-theoretic facts to gather intuitions on a problem from a young age.
– Gâteau-Gallois
Nov 21 at 17:08
"Taught in schools" ...which schools? :) Venn diagrams and partitions are surely "very useful" in any elementary problems (up to highschool level), but the more advanced stuff I'd think arent necessary as long as the sample space is finite. Once you run into infinite / continuous space you might need a good foundation, or else you end up wasting time arguing about "paradoxes" (conditioning on an event of prob 0, en.wikipedia.org/wiki/Bertrand_paradox_(probability), en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_paradox etc)
– antkam
Nov 21 at 20:28