Question

at sangers auto garage, three out of every five cars brought in for service need an oil change. of the cars that need an oil change, four out of every seven also need a tire rotation. what is the probability that a car that comes into the garage needs both an oil change and a tire rotation? give the answer in fraction form.

Explanation:

Step1: Define Probabilities

Let \( P(O) \) be the probability a car needs an oil change, so \( P(O)=\frac{3}{5} \). Let \( P(R|O) \) be the probability a car needs a tire rotation given it needs an oil change, so \( P(R|O)=\frac{4}{7} \).

Step2: Apply Multiplication Rule

The probability of both events (oil change and tire rotation) is \( P(O\cap R)=P(O)\times P(R|O) \). Substitute the values: \( \frac{3}{5}\times\frac{4}{7}=\frac{12}{35} \).

Answer:

\(\frac{12}{35}\)