EX 2.25 buying digital camera

• Suppose that of all individuals buying a certain digital camera,

  • $60\%$ include an optional memory card in their purchase,
  • $40\%$ include an extra battery,
  • and $30\%$ include both a card and battery.

• Consider randomly selecting a buyer and let

  • $A=\{$ memory card purchased $\}$
  • $B=\{$ battery purchased $\}$.

• Then

  • $P\left(A\right)=.60$ ,
  • $P\left(B\right)=.40$
  • $P\left(\text{both purchased}\right)=P\left(A\cap B\right)=.30$.

• Given that the selected individual purchased an extra battery,

  • the probability that an optional card was also purchased is
  • $P\left(A\mid B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}=\frac{.30}{.40}=.75$
  • That is, of all those purchasing an extra battery,
    • $75\%$ purchased an optional memory card.

• Similarly,

$$\begin{array}{rl}P\left(\text{battery}\mid \text{memory card}\right)& =P\left(B\mid A\right)\\ & =\frac{P\left(A\cap B\right)}{P\left(A\right)}=\frac{.30}{.60}=.50\end{array}$$

• Notice that
  • $P\left(A\mid B\right)\ne P\left(A\right)$
  • $P\left(B\mid A\right)\ne P\left(B\right)$ .