EX 2.34 washer vs dryer

• It is known that
  • $30\%$ of a certain company’s washing machines require service while under warranty,
  • whereas only $10\%$ of its dryers need such service.
• If someone purchases both a washer and a dryer made by this company,

Question

What is the probability that both machines will need warranty service?

• Let
  • $A$ denote the event that the washer needs service while under warranty,
  • $B$ be defined analogously for the dryer.
• Then
  • $P\left(A\right)=.30$
  • $P\left(B\right)=.10$.
• Assuming that the two machines will function independently of one another,
  • the desired probability is

$$P\left(A\cap B\right)=P\left(A\right)\cdot P\left(B\right)=\left(.30\right)\left(.10\right)=.03$$

• It is straightforward to show that $A$ and $B$ are independent
  • iff $A^{\prime }$ and $B$ are independent,
  • iff $A$ and $B^{\prime }$ are independent,
  • and iff $A^{\prime }$ and $B^{\prime }$ are independent.
• Thus in Example 2.34, the probability that neither machine needs service is

$$P\left(A^{\prime }\cap B^{\prime }\right)=P\left(A^{\prime }\right)\cdot P\left(B^{\prime }\right)=\left(.70\right)\left(.90\right)=.63$$