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: Find the probability of needing an oil change

The probability that a car needs an oil change is \(P(\text{oil change})=\frac{3}{5}\).

Step2: Find the conditional probability of needing a tire rotation given an oil change

The probability that a car needs a tire rotation given it needs an oil change is \(P(\text{tire rotation}|\text{oil change})=\frac{4}{7}\).

Step3: Use the formula for the probability of the intersection of two events

The formula for \(P(A\cap B)=P(A)\times P(B|A)\). Here \(A\) is "needing an oil change" and \(B\) is "needing a tire rotation". So \(P(\text{oil change}\cap\text{tire rotation})=\frac{3}{5}\times\frac{4}{7}\).
Calculate \(\frac{3}{5}\times\frac{4}{7}=\frac{3\times4}{5\times7}=\frac{12}{35}\).

Answer:

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