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. Given 3 out of 5 cars need an oil change, so \( P(O)=\frac{3}{5} \). Let \( P(T|O) \) be the probability a car needs a tire rotation given it needs an oil change. Given 4 out of 7 cars that need an oil change also need a tire rotation, so \( P(T|O)=\frac{4}{7} \).

Step2: Apply Conditional Probability Formula

The formula for conditional probability is \( P(T \cap O)=P(O)\times P(T|O) \). Substitute the values: \( P(T \cap O)=\frac{3}{5}\times\frac{4}{7} \).

Step3: Calculate the Product

Multiply the numerators and denominators: \( \frac{3\times4}{5\times7}=\frac{12}{35} \).

Answer:

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