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  • Conditional Probability and Independence

Conditional Probability

• $P(A|B)=$ the probability of event $A$ if we have the condition event $B$ already having occured
• $P(A|B)$
  • We know that $B$ has occured.
  • So the sample space shrinks
  • Now the sample space shrinks to $P(B)$
  • We can't do something in $A\setminus B$
    • Only an event can happen in $A\cap B$
    • So $P(A|B)=\frac{P(A\cap B)}{P(B)}$
  • Example:
    • Find the probability of finding a female undergraduate
      • Multiplication Law of Probability is used below
      • $P(F\cap L\cap u)=(0.6)(0.7)(0.5)$
      • $P(F\cap L\cap u)=0.21$
      • Or $P(F)\cdot P(L|F)\cdot P(u|F\cap L)$
    • Now if we want probability of female, local, undergraduate or female, local, PhD
      • We add these two.
      • $P(F\cap L\cap PhD)=(0.6)(0.7)(0.2)$
      • $P(F\cap L\cap PhD)=0.084$
      • $P(F\cap L\cap PhD)+P(F\cap L\cap u)=0.084+0.21$
      • $P(F\cap L\cap PhD)+P(F\cap L\cap u)=0.294$
      • We can just add because there's no overlap