Chapter 10.3, Problem 94E

(a)

To determine

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

Answer to Problem 94E

Thus, the probability is $0.0031$ .

Explanation of Solution

The binomial probability is as:

  $P(X=k)=C{}_k\times p^k\times {(1-p)}^{n-k}$

Now, it is given that:

  $\begin{array}{l}n=12\\ p=\frac{1}{2}=0.5\end{array}$

As this follows the binomial distribution then, evaluate it at $k=11,12$ :

  $\begin{array}{l}P(X=11)=C{}_{11}\times {(}^{0.5}\times {(}^1\\ =\frac{12!}{11!1!}\times {(}^{0.5}\times {(}^{0.5}\\ =0.0029\\ P(X=12)=C{}_{12}\times {(}^{0.5}\times {(}^1\\ =\frac{12!}{12!0!}\times {(}^{0.5}\times {(}^{0.5}\\ =0.0002\end{array}$

Now, we will use the addition rule for the following:

  $\begin{array}{l}P(X\ge 11)=P(X=11)+P(X=12)\\ =0.0029+0.0002\\ =0.0031\\ =0.31\%\end{array}$

(b)

To determine

To explain what conclusion would you make about the Emirates pilot’s suspicion based on the result of part (a).

Answer to Problem 94E

There is a convincing evidence that the Dubai-to-Doha outbound flights typically take longer.

Explanation of Solution

From part (a), we have the probability is $0.0031$ . And the pilot wants to know that the Dubai-to-Doha outbound flights typically take longer time or not.

Then according to the result in part (A) we conclude that the probability is small which indicate that it is unlikely to obtain at least $11\text{ of the 12}$ days had on outbound flight that took longer when the outbound flight and return flight are equally likely to have taken longer.

This implies that there is a convincing evidence that the Dubai-to-Doha outbound flights typically take longer.