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 classic car wash if they do not vacuum their car? 0.26 0.35 0.54 0.75

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

Answer

Answer:

0.75

Explanation:

Step1: Identify relevant probabilities

We want $P(\text{Classic}|\text{No Vacuum})$. By Bayes' theorem and total - probability formula, first find $P(\text{No Vacuum})$. $P(\text{No Vacuum})=0.45\times0.90 + 0.35\times(1 - 0.25)+0.20\times(1 - 0.60)$. $P(\text{No Vacuum})=0.45\times0.90+0.35\times0.75 + 0.20\times0.40$. $P(\text{No Vacuum})=0.405+0.2625 + 0.08=0.7475$. $P(\text{Classic and No Vacuum})=0.35\times0.75 = 0.2625$.

Step2: Apply conditional - probability formula

$P(\text{Classic}|\text{No Vacuum})=\frac{P(\text{Classic and No Vacuum})}{P(\text{No Vacuum})}=\frac{0.35\times0.75}{0.45\times0.90 + 0.35\times0.75+0.20\times0.40}=\frac{0.2625}{0.7475}\approx0.75$.