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 basic or classic car wash if they vacuum their car? 0.13 0.48 0.52 0.80

Explanation:

Step1: Calculate the probability of basic wash and vacuum

The probability of a customer getting a basic wash is \(0.45\), and the probability of vacuuming given a basic wash is \(0.10\). Using the multiplication rule for dependent events \(P(A\cap B)=P(A)\times P(B|A)\), we have \(P(\text{Basic}\cap\text{Vacuum}) = 0.45\times0.10=0.045\).

Step2: Calculate the probability of classic wash and vacuum

The probability of a customer getting a classic wash is \(0.35\), and the probability of vacuuming given a classic wash is \(0.25\). Using the multiplication rule \(P(\text{Classic}\cap\text{Vacuum})=0.35\times0.25 = 0.0875\).

Step3: Calculate the probability of ultimate wash and vacuum

The probability of a customer getting an ultimate wash is \(0.20\), and the probability of vacuuming given an ultimate wash is \(0.60\). So \(P(\text{Ultimate}\cap\text{Vacuum})=0.20\times0.60 = 0.12\).

Step4: Use the formula for conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here \(A\) is the event of getting a basic or classic wash, and \(B\) is the event of vacuuming. \(P((\text{Basic}\cup\text{Classic})\cap\text{Vacuum})=P(\text{Basic}\cap\text{Vacuum})+P(\text{Classic}\cap\text{Vacuum})=0.045 + 0.0875=0.1325\), and \(P(\text{Vacuum})=P(\text{Basic}\cap\text{Vacuum})+P(\text{Classic}\cap\text{Vacuum})+P(\text{Ultimate}\cap\text{Vacuum})=0.045+0.0875 + 0.12=0.2525\). Then \(P((\text{Basic}\cup\text{Classic})|\text{Vacuum})=\frac{0.045 + 0.0875}{0.045+0.0875 + 0.12}=\frac{0.1325}{0.2525}=\frac{1325}{2525}=\frac{53}{101}\approx0.52\)

Answer:

\(0.52\)