Question

: Recall the airplane cargo problem we have discussed in our first lecture. An air-freight company has 8 adjacent positions on its Boeing-727 aircraft for freight containers. The weights of this containers depend on what they are carrying. and company statistics indicate that %7 of the containers are classified as ”heavy”. While heavy containers are not inherently dangerous, having two such containers next to each other is considered dangerous should the plane encounter a wind gust.

Understandably, company wants to know how often will it be in a situation where two heavy cargo are next to each other, given that it loads the cargo in a random fashion. One of the engineers claim that this scenario is very unlikely, since chances of loading a single heavy crate is low (%7) and hence having two such containers next to each other is a very low chance event. Is the engineer right? Discuss using random variables and probabilities.

Answer 1

We can model the situation using Binomial distribution as follows.

Suppose, random variable X denotes number of heavy containers.

Being a container heavy or not is independent of other containers.

Denoting having a heavy container as success we get,

Two heavy containers can be placed beside only when at least 2 heavy containers in an aircraft.

So, probability of such situation is very small.

Further we observe that , which is significantly small.

We observe that

So, in case of X=2, we calculate probability of two containers to be placed beside.

Possible number of all types of arrangements is

Possible number of arrangements in which 2 heavy containers are placed beside is

So, probability of such arrangement given that there are 2 heavy containers is

Hence, probability that there are 2 heavy containers and those are placed beside

Clearly, we observe that this value is very low.

Since, probability of having more than 2 heavy containers in an aircraft is very small, those occur in very rare cases.

Hence, we can conclude that having two such containers next to each other is a very low chance event.