EX 2.25

EX 2.25

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

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

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