问题

For customers purchasing a full set of tires at a particular tire store, consider the events
\[
A=\{\text{tires purchased were made in the United States}\}
\]
\[
B=\{\text{purchaser has tires balanced immediately}\}
\]
\[
C=\{\text{purchaser requests front-end alignment}\}
\]
along with $A^c$, $B^c$, and $C^c$. Assume the following unconditional and conditional probabilities:
\[
P(A)=.75\qquad P(B|A)=.9\qquad P(B|A^c)=.8
\]
\[
P(C|A\cap B)=.8\qquad P(C|A\cap B^c)=.6
\]
\[
P(C|A^c\cap B)=.7\qquad P(C|A^c\cap B^c)=.3
\]

1.  Construct a tree diagram consisting of first-, second-, and third-generation branches and place an event label and appropriate probability next to each branch.
2.  Compute $P(A\cap B\cap C)$.
3.  Compute $P(B\cap C)$.
4.  Compute $P(C)$.
5.  Compute $P(A|B\cap C)$, the probability of a purchase of U.S. tires given that both balancing and an alignment were requested.