Question

On a used car lot, $50\%$ of the vehicles are cars, $\dfrac{3}{4}$ of which have automatic transmissions, $\dfrac{1}{3}$ have leather interiors. If a vehicle is chosen from the lot at random, what is the probability that it will be a car with an automatic transmission and a leather interior?
$A)\dfrac{1}{8}$
$B)\dfrac{1}{6}$
$C)\dfrac{1}{4}$
$D)\dfrac{1}{3}$

Answer 1

First we have to define what the terms we need to solve the problem are.
Since exactly fifty percent of the vehicles are cars on the car lot, some have automatic transmission with the probability of $\dfrac{3}{4}$ and some cars have leather interiors with the probability of $\dfrac{1}{3}$. Suppose the vehicle is chosen in the lot at random (can be anything), we need to find the probability of a car with an automatic transmission and a leather interior.
Formula used: $\text{Probability} = \dfrac{\text{Favorable event}}{\text{Total event}}$

Complete step-by-step solution:
Since we need to find the probability of car with an automatic transmission and a leather interior; let the total number of vehicles fixed as $N$, and here fifty percentage are cars only and hence we get the total cars are at $\dfrac{N}{2}$. Now we are going to multiply the total number of cars into the automatic transmission with the probability of $\dfrac{3}{4}$; we get $\dfrac{N}{2} \times \dfrac{3}{4} = \dfrac{{3N}}{8}$ but since the total number of the cars having leather interiors too and hence we get $\dfrac{1}{3} \times \dfrac{{3N}}{8} = \dfrac{N}{8}$ (multiplied with total cars), hence this is the favorable events and the total events are $N$ (total vehicles). Thus, using the formula, we get;
$\text{Probability} = \dfrac{\text{Favorable event}}{\text{Total event}}$=
$\text{Probability} = \dfrac{{\dfrac{N}{8}}}{N}{\text{ }}$ which can be written as in the form of $\text{Probability} = \dfrac{{\dfrac{N}{8}}}{N}{\text{ }} = \dfrac{1}{8}$ (common terms N will be canceled)
Thus, we get option $A)\dfrac{1}{8}$ is correct

Note: if they ask us to find the percentage of a car with an automatic transmission and a leather interior; we need to use the percentage formula which is $\dfrac{1}{8} \times 100$ and further solving we get, $\dfrac{1}{8} \times 100 \Rightarrow 12.5$ (is the percentage of getting cars with an automatic transmission and a leather interior.