Probability theory MOC

Conditional probability

Conditional probability allows the investigation of how the knowledge of one event occurring effects the knowledge of the other one. Given a probability model (πœ‰,F,β„™), and two events 𝐴,𝐡 ∈F, the conditional probability of 𝐴 given 𝐡 is prob

β„™(𝐴∣𝐡)=β„™(𝐴∩𝐡)β„™(𝐡)

unless β„™(𝐡) =0, in which case β„™(𝐴 ∣𝐡) =0.1 The function β„™( β‹… ∣𝐴) forms a probability measure on the same space (πœ‰,F) as β„™.

Properties

β„™(𝐴∩𝐡)=β„™(𝐡)β„™(𝐴∣𝐡)=β„™(𝐴)β„™(𝐡∣𝐴)
β„™(𝐴∣𝐡)=β„™(𝐡∣𝐴)β„™(𝐴)β„™(𝐡)
β„™(𝐴∣𝐡)β„™(π΄π‘βˆ£π΅)=β„™(𝐡∣𝐴)β„™(π΅βˆ£π΄π‘)β„™(𝐴)β„™(𝐴𝑐)
  1. Let {𝐴𝑖}𝑛𝑖=1 partition πœ‰. Then
β„™(𝐡)=π‘›βˆ‘π‘–=1β„™(π΅βˆ£π΄π‘–)β„™(𝐴𝑖)
β„™(𝐴∣𝐡∩𝐸)=β„™(𝐡∣𝐴∩𝐸)β„™(𝐴∣𝐸)β„™(𝐡∣𝐸)

See also

tidy | SemBr | en

Footnotes

  1. Since this may be considered an impossible scenario. ↩

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