Question

an automobile manufacturing plant produced 32 vehicles today: 14 were trucks, 7 were vans, and 11 were motorcycles. (each vehicle falls into only one of these categories.) plant managers are going to select two of these vehicles for a thorough inspection. the first vehicle will be selected at random, and then the second vehicle will be selected at random from the remaining vehicles. what is the probability that two vans will be selected? do not round your intermediate computations. round your final answer to three decimal places.

Explanation:

Step1: Calculate probability of first - van selection

The total number of vehicles is 32 and the number of vans is 7. The probability of selecting a van as the first vehicle, $P(V_1)$, is $\frac{7}{32}$.

Step2: Calculate probability of second - van selection

After one van is selected, there are 31 vehicles left and 6 vans left. The probability of selecting a van as the second vehicle given that the first one was a van, $P(V_2|V_1)$, is $\frac{6}{31}$.

Step3: Calculate the probability of both vans

By the multiplication rule for dependent events, $P(V_1\cap V_2)=P(V_1)\times P(V_2|V_1)$. So $P(V_1\cap V_2)=\frac{7}{32}\times\frac{6}{31}=\frac{42}{992}\approx0.042$.

Answer:

0.042