Quest for a non-inductive proof of the addition theorem of probability.
The addition theorem of probability states that:
$$P(A_1cup A_2cup … cup A_n) = sum P(A_i) - sum P(A_icap A_j) + sum P(A_icap A_jcap A_k) - … + (-1)^nP(A_1cap A_2cap … cap A_n)$$
Although this result is proved using the method of induction (as in the case of most textbooks), can it be proved directly?
The only thing which I see is its structural similarity with the general product of binomials :
$$(1+alpha)(1+beta)……(1+nu) = 1 + (alpha + beta + … + nu) + (alpha beta + ……) + ……… + (alpha beta gamma …nu)$$
However I am unable to get a link between the two. Any help is appreciated.
probability
add a comment |
The addition theorem of probability states that:
$$P(A_1cup A_2cup … cup A_n) = sum P(A_i) - sum P(A_icap A_j) + sum P(A_icap A_jcap A_k) - … + (-1)^nP(A_1cap A_2cap … cap A_n)$$
Although this result is proved using the method of induction (as in the case of most textbooks), can it be proved directly?
The only thing which I see is its structural similarity with the general product of binomials :
$$(1+alpha)(1+beta)……(1+nu) = 1 + (alpha + beta + … + nu) + (alpha beta + ……) + ……… + (alpha beta gamma …nu)$$
However I am unable to get a link between the two. Any help is appreciated.
probability
You could decompose the union into disjoint sets. For instance, $A cup B = (A cap B) cup (A cap B^c) cup (A^c cap B)$. Now rearrange $P(A) = P(A cap B) + P(A cap B^c)$.
– Stockfish
Nov 16 '18 at 12:46
Of course, that's not really an answer to your question ...
– Stockfish
Nov 16 '18 at 12:53
@Stockfish, I think it may involve polynomial fields and other concepts of linear algebra, when we are looking for the decomposition of the general union operator and further rearrangement.
– Awe Kumar Jha
Nov 16 '18 at 12:58
@Tusky, Ah, that was very helpful.
– Awe Kumar Jha
Nov 16 '18 at 13:20
add a comment |
The addition theorem of probability states that:
$$P(A_1cup A_2cup … cup A_n) = sum P(A_i) - sum P(A_icap A_j) + sum P(A_icap A_jcap A_k) - … + (-1)^nP(A_1cap A_2cap … cap A_n)$$
Although this result is proved using the method of induction (as in the case of most textbooks), can it be proved directly?
The only thing which I see is its structural similarity with the general product of binomials :
$$(1+alpha)(1+beta)……(1+nu) = 1 + (alpha + beta + … + nu) + (alpha beta + ……) + ……… + (alpha beta gamma …nu)$$
However I am unable to get a link between the two. Any help is appreciated.
probability
The addition theorem of probability states that:
$$P(A_1cup A_2cup … cup A_n) = sum P(A_i) - sum P(A_icap A_j) + sum P(A_icap A_jcap A_k) - … + (-1)^nP(A_1cap A_2cap … cap A_n)$$
Although this result is proved using the method of induction (as in the case of most textbooks), can it be proved directly?
The only thing which I see is its structural similarity with the general product of binomials :
$$(1+alpha)(1+beta)……(1+nu) = 1 + (alpha + beta + … + nu) + (alpha beta + ……) + ……… + (alpha beta gamma …nu)$$
However I am unable to get a link between the two. Any help is appreciated.
probability
probability
edited Nov 16 '18 at 13:15
Asaf Karagila♦
302k32426756
302k32426756
asked Nov 16 '18 at 12:42
Awe Kumar Jha
38613
38613
You could decompose the union into disjoint sets. For instance, $A cup B = (A cap B) cup (A cap B^c) cup (A^c cap B)$. Now rearrange $P(A) = P(A cap B) + P(A cap B^c)$.
– Stockfish
Nov 16 '18 at 12:46
Of course, that's not really an answer to your question ...
– Stockfish
Nov 16 '18 at 12:53
@Stockfish, I think it may involve polynomial fields and other concepts of linear algebra, when we are looking for the decomposition of the general union operator and further rearrangement.
– Awe Kumar Jha
Nov 16 '18 at 12:58
@Tusky, Ah, that was very helpful.
– Awe Kumar Jha
Nov 16 '18 at 13:20
add a comment |
You could decompose the union into disjoint sets. For instance, $A cup B = (A cap B) cup (A cap B^c) cup (A^c cap B)$. Now rearrange $P(A) = P(A cap B) + P(A cap B^c)$.
– Stockfish
Nov 16 '18 at 12:46
Of course, that's not really an answer to your question ...
– Stockfish
Nov 16 '18 at 12:53
@Stockfish, I think it may involve polynomial fields and other concepts of linear algebra, when we are looking for the decomposition of the general union operator and further rearrangement.
– Awe Kumar Jha
Nov 16 '18 at 12:58
@Tusky, Ah, that was very helpful.
– Awe Kumar Jha
Nov 16 '18 at 13:20
You could decompose the union into disjoint sets. For instance, $A cup B = (A cap B) cup (A cap B^c) cup (A^c cap B)$. Now rearrange $P(A) = P(A cap B) + P(A cap B^c)$.
– Stockfish
Nov 16 '18 at 12:46
You could decompose the union into disjoint sets. For instance, $A cup B = (A cap B) cup (A cap B^c) cup (A^c cap B)$. Now rearrange $P(A) = P(A cap B) + P(A cap B^c)$.
– Stockfish
Nov 16 '18 at 12:46
Of course, that's not really an answer to your question ...
– Stockfish
Nov 16 '18 at 12:53
Of course, that's not really an answer to your question ...
– Stockfish
Nov 16 '18 at 12:53
@Stockfish, I think it may involve polynomial fields and other concepts of linear algebra, when we are looking for the decomposition of the general union operator and further rearrangement.
– Awe Kumar Jha
Nov 16 '18 at 12:58
@Stockfish, I think it may involve polynomial fields and other concepts of linear algebra, when we are looking for the decomposition of the general union operator and further rearrangement.
– Awe Kumar Jha
Nov 16 '18 at 12:58
@Tusky, Ah, that was very helpful.
– Awe Kumar Jha
Nov 16 '18 at 13:20
@Tusky, Ah, that was very helpful.
– Awe Kumar Jha
Nov 16 '18 at 13:20
add a comment |
1 Answer
1
active
oldest
votes
Your very statement:
$$ P(A1∪A2∪…∪An)=∑P(Ai)−∑P(Ai∩Aj)+∑P(Ai∩Aj∩Ak)−…+(−1)nP(A1∩A2∩…∩An) $$
is what we arrive at from what we call the Inclusion-Exclusion Principle.
Since you are looking for a non-induction based proof, there exists a proof using the Binomial Theorem here and an algebraic proof using indicator functions over here.
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3001097%2fquest-for-a-non-inductive-proof-of-the-addition-theorem-of-probability%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Your very statement:
$$ P(A1∪A2∪…∪An)=∑P(Ai)−∑P(Ai∩Aj)+∑P(Ai∩Aj∩Ak)−…+(−1)nP(A1∩A2∩…∩An) $$
is what we arrive at from what we call the Inclusion-Exclusion Principle.
Since you are looking for a non-induction based proof, there exists a proof using the Binomial Theorem here and an algebraic proof using indicator functions over here.
add a comment |
Your very statement:
$$ P(A1∪A2∪…∪An)=∑P(Ai)−∑P(Ai∩Aj)+∑P(Ai∩Aj∩Ak)−…+(−1)nP(A1∩A2∩…∩An) $$
is what we arrive at from what we call the Inclusion-Exclusion Principle.
Since you are looking for a non-induction based proof, there exists a proof using the Binomial Theorem here and an algebraic proof using indicator functions over here.
add a comment |
Your very statement:
$$ P(A1∪A2∪…∪An)=∑P(Ai)−∑P(Ai∩Aj)+∑P(Ai∩Aj∩Ak)−…+(−1)nP(A1∩A2∩…∩An) $$
is what we arrive at from what we call the Inclusion-Exclusion Principle.
Since you are looking for a non-induction based proof, there exists a proof using the Binomial Theorem here and an algebraic proof using indicator functions over here.
Your very statement:
$$ P(A1∪A2∪…∪An)=∑P(Ai)−∑P(Ai∩Aj)+∑P(Ai∩Aj∩Ak)−…+(−1)nP(A1∩A2∩…∩An) $$
is what we arrive at from what we call the Inclusion-Exclusion Principle.
Since you are looking for a non-induction based proof, there exists a proof using the Binomial Theorem here and an algebraic proof using indicator functions over here.
edited yesterday
answered Nov 16 '18 at 13:34
Tusky
644618
644618
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3001097%2fquest-for-a-non-inductive-proof-of-the-addition-theorem-of-probability%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
You could decompose the union into disjoint sets. For instance, $A cup B = (A cap B) cup (A cap B^c) cup (A^c cap B)$. Now rearrange $P(A) = P(A cap B) + P(A cap B^c)$.
– Stockfish
Nov 16 '18 at 12:46
Of course, that's not really an answer to your question ...
– Stockfish
Nov 16 '18 at 12:53
@Stockfish, I think it may involve polynomial fields and other concepts of linear algebra, when we are looking for the decomposition of the general union operator and further rearrangement.
– Awe Kumar Jha
Nov 16 '18 at 12:58
@Tusky, Ah, that was very helpful.
– Awe Kumar Jha
Nov 16 '18 at 13:20