Question

Boeing 787 Dreamliner. An assessment of the new Boeing 787 “Dreamliner” airplane was published in the Journal of Case Research in Business and Economics (December 2014). Boeing planned to build and deliver 3,500 of these aircraft within the first 20 years of the 787 program. The researchers used simulation to estimate the number of annual deliveries of 787 airplanes (expressed as a percentage of total Boeing aircraft deliveries during the year). In the fifth year of the program, deliveries were uniformly distributed over the range 2.5% to 5.5%. Assuming the simulated results are accurate, what is the probability that the number of Boeing 787s built and delivered in the fifth year of the program exceeds 4% of the total Boeing aircraft deliveries during the year?

Short answer

The probability that the number of Boeing 787s built and delivered in the program's fifth year exceeds 4% of the total Boeing aircraft deliveries during the year is 0.5.

Step-by-step solution

Given information

The deliveries of 787 Dreamliner airplanes in the fifth year of the 787 programs are expressed as a percentage of total Boeing aircraft deliveries during the year.

Define the random variable and compute the probability

Let X denotes the percentage of Boeing 787s delivered in the fifth year of the program of the total Boeing aircraft.

Therefore, X is uniformly distributed in the interval between 2.5 and 5.5.

The probability that the number of Boeing 787s built and delivered in the fifth year of the program exceeds 4% of the total Boeing aircraft deliveries during the year is obtained as:

$\begin{array}{rcl}P\left(X>4\right)& =& P\left(4<X<5.5\right)\\ & =& \frac{5.5-4}{5.5-2.5}\\ & =& \frac{1.5}{3}\\ & =& 0.5\end{array}$

For the uniform distribution: $\mathit{P}\left(\mathit{a}<\mathit{x}<\mathit{b}\right)\mathbf{=}\frac{\mathbf{b}\mathbf{-}\mathbf{a}}{\mathbf{d}\mathbf{-}\mathbf{c}}\mathbf{,}\mathit{c}\mathbf{⩽}\mathit{a}\mathbf{<}\mathit{b}\mathbf{⩽}\mathit{d}$.

Therefore, the required probability is 0.5.