Is the Formula for the Probability of Symmetric Difference Accurate?

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Homework Help Overview

The discussion revolves around the formula for the probability of the symmetric difference of two sets, specifically P(AΔB) = P(A) + P(B) - 2P(A∩B). Participants are tasked with demonstrating the accuracy of this formula in a probability context.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants attempt to manipulate the formula and express P(AΔB) in terms of intersections and unions of sets. There is a question regarding what it means to "show" that the formula is accurate, prompting further clarification on the task. Some suggest using Venn diagrams as a visual aid.

Discussion Status

The discussion is ongoing, with participants sharing their attempts and hints. There is no explicit consensus on the next steps, but hints and suggestions for visual aids have been provided to guide the exploration.

Contextual Notes

Some participants express uncertainty about the requirements of the task, particularly regarding the definition and implications of the symmetric difference in probability. There is also a repeated emphasis on the formula itself without additional context or definitions provided.

GreenPrint
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Homework Statement



Show that the formula: P(AΔB) = P(A) + P(B) - 2P(A[itex]\bigcap[/itex]B)

Homework Equations





The Attempt at a Solution



P(AΔB) = P(A) + P(B) - 2P(A[itex]\bigcap[/itex]B)

P(AΔB) = P(A[itex]\bigcap B^{c}[/itex])[itex]\bigcup (A^{c}\bigcap B)[/itex]

I don't know where to go from here. Thanks for any help.
 
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GreenPrint said:

Homework Statement



Show that the formula: P(AΔB) = P(A) + P(B) - 2P(A[itex]\bigcap[/itex]B)
Show that the formula does what?
GreenPrint said:

Homework Equations

Definition of P(AΔB), perhaps?
GreenPrint said:

The Attempt at a Solution



P(AΔB) = P(A) + P(B) - 2P(A[itex]\bigcap[/itex]B)

P(AΔB) = P(A[itex]\bigcap B^{c}[/itex])[itex]\bigcup (A^{c}\bigcap B)[/itex]

I don't know where to go from here. Thanks for any help.
 
Hi GreenPrint! :smile:

Hint: how would you prove area(AΔB) = area(A) + area(B) - 2area(A[itex]\bigcap[/itex]B) ? :wink:
 
GreenPrint said:

Homework Statement



Show that the formula: P(AΔB) = P(A) + P(B) - 2P(A[itex]\bigcap[/itex]B)

Homework Equations





The Attempt at a Solution



P(AΔB) = P(A) + P(B) - 2P(A[itex]\bigcap[/itex]B)

P(AΔB) = P(A[itex]\bigcap B^{c}[/itex])[itex]\bigcup (A^{c}\bigcap B)[/itex]

I don't know where to go from here. Thanks for any help.

Use Venn diagrams; that's their purpose.
 

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