Flight times The pilot suspects that the Dubai-to-Doha outbound flight typically takes longer. If so, the difference (Outbound−Return) in flight times will be positive on more days than it is negative. What if either flight is equally likely to take longer? Then we can model the outcome on a randomly selected day with a coin toss. Heads means the Dubai-to-Doha outbound flight lasts longer; tails means the Doha-to-Dubai return flight lasts longer. To imitate a random sample of $12$days, imagine tossing a fair coin $12$times.

a. Find the probability of getting $11$or more heads in $12$tosses of a fair coin.

b. The outbound flight took longer on $11$of the $12$days. Based on your result in part (a), what conclusion would you make about the Emirates pilot’s suspicion?

We need to find probability of getting $11$or more heads in $12$ times.

As we know, the bionomial probability is :

$$\begin{gathered} P(X=k)={}^{n}C_k\times p^k\times {(1-p)}^{n-k}n=12 \\ p=0.5 \\ P(X=11)={}^{12}C_{11}\times 0.5^{11}\times {(1-0.5)}^{12-11} \\ =0.0029 \\ P(X=12)={}^{12}C_{12}\times 0.5^{12}\times {(1-0.5)}^{12-12} \\ =0.0002 \end{gathered}$$

Using, addition rule,

$$\begin{gathered} \mathrm{P}(\mathrm{X}\ge 11)=\mathrm{P}(\mathrm{X}=11)+\mathrm{P}(\mathrm{X}=12) \\ =0.0029+0.0002 \\ =0.31\% \end{gathered}$$

We need to conclude that what to make about the Emirates pilot’s suspicion.

The chance is $0.0031$, according to section (a).
And the pilot wants to know if outbound flights from Dubai to Doha are normally lengthier.
Then, based on the finding in part (a), we conclude that the likelihood is minimal, implying that at least $11$of the $12$ days were spent on an outward journey that took longer when both the outbound and return flights were equally likely to take longer.
This suggests that there is strong evidence that outbound flights from Dubai to Doha take longer on average.