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: Calculate the probability of needing an oil change

The probability of a car needing an oil change is \(\frac{3}{5}\) since three out of every five cars need an oil change.

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

The probability of a car needing a tire rotation given that it needs an oil change is \(\frac{4}{7}\) (four out of every seven cars that need an oil change also need a tire rotation).

Step3: Use the multiplication rule for probability

The multiplication rule for probability of two - dependent events \(A\) (needing an oil change) and \(B\) (needing a tire rotation) is \(P(A\cap B)=P(A)\times P(B|A)\).
Substitute \(P(A)=\frac{3}{5}\) and \(P(B|A)=\frac{4}{7}\) into the formula:
\(P(A\cap B)=\frac{3}{5}\times\frac{4}{7}=\frac{3\times4}{5\times7}=\frac{12}{35}\)

Answer:

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