Whats the most insightful probability post ever posted on here?

["spaceman"Bryce]

it would be interesting to see....

[elitegimp]

I really liked BruceZ's inclusion-exlusion post... It might not be the most insightful, but I hope that by mentioning it he'll appear like a magic genie and post a link [img]/images/graemlins/smile.gif[/img] (I'm too lazy to search, and too lazy to PM him, so I guess if I never see the post again it'll be my own damn fault)

[jason_t]

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I really liked BruceZ's inclusion-exlusion post... It might not be the most insightful, but I hope that by mentioning it he'll appear like a magic genie and post a link [img]/images/graemlins/smile.gif[/img] (I'm too lazy to search, and too lazy to PM him, so I guess if I never see the post again it'll be my own damn fault)

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The second most beautiful counting method in combinatorics: Inclusion-Exclusion

[gaming_mouse]

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The second most beautiful counting method in combinatorics: Inclusion-Exclusion

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The first being?

[elitegimp]

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I really liked BruceZ's inclusion-exlusion post... It might not be the most insightful, but I hope that by mentioning it he'll appear like a magic genie and post a link [img]/images/graemlins/smile.gif[/img] (I'm too lazy to search, and too lazy to PM him, so I guess if I never see the post again it'll be my own damn fault)

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The second most beautiful counting method in combinatorics: Inclusion-Exclusion

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Awesome dude, thanks!

[jason_t]

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The second most beautiful counting method in combinatorics: Inclusion-Exclusion

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The first being?

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The method of double counting.

[gaming_mouse]

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The second most beautiful counting method in combinatorics: Inclusion-Exclusion

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The first being?

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The method of double counting.

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What's the difference?

[PygmyHero]

You lost me. Could we maybe have a link or thread for that too? Thanks!

[jason_t]

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The second most beautiful counting method in combinatorics: Inclusion-Exclusion

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The first being?

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The method of double counting.

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What's the difference?

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Double counting means enumerating a set by counting the objects in the set in two different ways and then equating them to arrive at the answer.

Inclusion-exclusion is the method of enumerating a set by overcounting and undercounting repeatedly to arrive at the answer.

[BruceZ]

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The second most beautiful counting method in combinatorics: Inclusion-Exclusion

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The first being?

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The method of double counting.

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What's the difference?

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Double counting means enumerating a set by counting the objects in the set in two different ways and then equating them to arrive at the answer.

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Example: Compute the sum S = 1 + 2 + 3 + ... + 100.

S = 100 + 99 + 98 + ... + 1

Adding the above two equations gives:

2S = 101 + 101 + 101 + ...(100 times)

2S = 101*100 = 10,100

S = 5,050


Example: Compute the sum S = 1 + 1/2 + 1/4 + 1/8 + ...

(1/2)*S = 1/2 + 1/4 + 1/8 + ...

Subtracting these gives:

(1/2)*S = 1

S = 2.


Example: S = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...

S = 1 + 1/S

S^2 = S + 1 = 0

S^2 - S - 1 = 0

S = 1/2 + sqrt(5)/2


Example: S = sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + ...

S = sqrt(1 + S)

S^2 = 1 + S

S^2 - S - 1 = 0

S = 1/2 + sqrt(5)/2


A well-known mathematician once told me that there are 4 tricks used to derive everything in applied mathematics (or something like that):

1. Exchange the order of integration or summation.
2. Integrate by parts.
3. Add and subtract 1.
4. Induction.