91Ó°ÊÓ

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

As we know, the bionomial probability is :

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

Using, addition rule,

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

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.