Question

a car wash has three different types of washes: basic, classic, and ultimate. based on records, 45% of customers get the basic wash, 35% get the classic wash, and 20% get the ultimate wash. some customers also vacuum out their cars after the wash. the car wash records show that 10% of customers who get the basic wash, 25% of customers who get the classic wash, and 60% of customers who get the ultimate wash also vacuum their cars. the probabilities are displayed in the tree diagram. what is the probability that a randomly selected customer purchases the classic car wash if they do not vacuum their car? 0.26 0.35 0.54 0.75

Explanation:

Step1: Calculate the probability of no - vacuum for each wash type

• For basic wash: \(P(\text{Basic and No Vacuum})=0.45\times0.90 = 0.405\)
• For classic wash: \(P(\text{Classic and No Vacuum})=0.35\times(1 - 0.25)=0.35\times0.75=0.2625\)
• For ultimate wash: \(P(\text{Ultimate and No Vacuum})=0.20\times(1 - 0.60)=0.20\times0.40 = 0.08\)

Step2: Calculate the total probability of no - vacuum

\(P(\text{No Vacuum})=0.405+0.2625 + 0.08=0.7475\)

Step3: Calculate the conditional probability

Using the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), where \(A\) is "Classic wash" and \(B\) is "No Vacuum"
\(P(\text{Classic}|\text{No Vacuum})=\frac{P(\text{Classic and No Vacuum})}{P(\text{No Vacuum})}=\frac{0.2625}{0.7475}\approx0.35\)

Answer:

\(0.35\)