EX 2.32 pumps in gas stations

• Consider
  • a gas station with six pumps numbered $1,2,\dots ,6$ ,
  • let $E_i$ denote the simple event that a randomly selected customer uses pump $i\left(i=1,\dots ,6\right)$.
• Suppose that
  • $P\left(E_1\right)=P\left(E_6\right)=.10$
  • $P\left(E_2\right)=P\left(E_5\right)=.15$
  • $P\left(E_3\right)=P\left(E_4\right)=.25$
• Define events $A,B,C$ by
  • $A=\{2,4,6\}$
  • $B=\{1,2,3\}$
  • $C=\{2,3,4,5\}$
• We then have
  • $P\left(A\right)=.50$,
  • $P\left(A\mid B\right)=.30$ ,
  • $P\left(A\mid C\right)=.50$ .
• That is,
  • events $A$ and $B$ are dependent,
  • whereas events $A$ and $C$ are independent.
• Intuitively, $A$ and $C$ are independent
  • because the relative division of probability among even- and odd-numbered pumps is the same among pumps $2,3,4,5$ as it is among all six pumps.