calculating a probability\nat sanger’s auto garage, three out of every five cars brought in for service need…

calculating a probability\nat sanger’s 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.\nwhat 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.

calculating a probability\nat sanger’s 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.\nwhat 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.

Answer

Explanation:

Step1: Find probability of oil - change

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

Step2: Find conditional probability of tire - rotation given oil - change

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

Step3: Use multiplication rule for conditional probability

By the multiplication rule $P(A\cap B)=P(A)\times P(B|A)$. Here, $A$ is the event of needing an oil - change and $B$ is the event of needing a tire - rotation. So $P(\text{oil change}\cap\text{tire rotation}) = P(\text{oil change})\times P(\text{tire rotation}|\text{oil change})=\frac{3}{5}\times\frac{4}{7}$.

Step4: Calculate the product

$\frac{3}{5}\times\frac{4}{7}=\frac{3\times4}{5\times7}=\frac{12}{35}$.

Answer:

$\frac{12}{35}$