a car wash has three different types of washes: basic, classic, and ultimate. based on records, 45% of…

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 ultimate car wash if they vacuum their car? 0.12 0.20 0.32 0.48

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 ultimate car wash if they vacuum their car? 0.12 0.20 0.32 0.48

Answer

Explanation:

Step1: Calculate the probability of getting each wash and vacuuming

  • Probability of basic wash and vacuum: $0.45\times0.10 = 0.045$.
  • Probability of classic wash and vacuum: $0.35\times0.25= 0.0875$.
  • Probability of ultimate wash and vacuum: $0.20\times0.60 = 0.12$.

Step2: Calculate the total probability of vacuuming

Total probability of vacuuming is the sum of the above - mentioned probabilities: $0.045 + 0.0875+0.12=0.2525$.

Step3: Use conditional - probability formula

The conditional - probability formula is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Here, $A$ is the event of getting the ultimate wash and $B$ is the event of vacuuming. $P(\text{ultimate}|\text{vacuum})=\frac{P(\text{ultimate}\cap\text{vacuum})}{P(\text{vacuum})}=\frac{0.12}{0.045 + 0.0875+0.12}=\frac{0.12}{0.2525}\approx0.48$.

Answer:

0.48