Languages in Canada Canada has two official languages, English and French. Choose a Canadian at random and ask, “What is your mother tongue?” Here is the distribution of responses, combining many separate languages from the broad Asia/Pacific region

a. Explain why this is a valid probability model.

b. What is the probability that the chosen person’s mother tongue is not English?

c. What is the probability that the chosen person’s mother tongue is one of Canada’s official languages?

We are given a table. We need to explain validity of probability model.

As we know sum of all probabilities must be $1$and probabilities should lie between $0$and $1$

Sum of probabilities is $0.63+0.22+0.06+0.09=1$

Now, from the table we can say it follows both above mentioned conditions.

So, We can say the probability model is valid.

We are given a table. We need to find the probability that the chosen person’s mother tongue is not English.

Using complement rule, $\mathrm{P}\left({\mathrm{A}}^{\mathrm{c}}\right)=\mathrm{P}\left(\overline{\mathrm{A}}\right)=1-\mathrm{P}\left(\mathrm{A}\right)$

Probability of English : $\mathrm{P}\left(\mathrm{E}\right)=0.63$

Applying complement rule,

$\mathrm{P}\left({\mathrm{E}}^{\mathrm{c}}\right)=1-\mathrm{P}\left(\mathrm{E}\right)=1-0.63=0.37$

Hence, $0.37$ is the probability that the chosen person’s mother tongue is not English.

We are given a table. We need to find the probability that the chosen person’s mother tongue is one of Canada’s official languages.

Using addition rule of mutually exclusive for disjoint events,

$\mathrm{P}(\mathrm{A}\mathrm{U}\mathrm{B})=\mathrm{P}(\mathrm{A}\mathrm{or}\mathrm{B})=\mathrm{P}\left(\mathrm{A}\right)+\mathrm{P}\left(\mathrm{B}\right)$

Probability of English $\mathrm{P}\left(\mathrm{E}\right)=0.63$

Probability of French $\mathrm{P}\left(\mathrm{F}\right)=0.22$

Now,

$$\begin{gathered} \mathrm{P}(\mathrm{E}\mathrm{U}\mathrm{F})=\mathrm{P}\left(\mathrm{E}\right)+\mathrm{P}\left(\mathrm{F}\right) \\ \mathrm{P}(\mathrm{E}\mathrm{U}\mathrm{F})=0.63+0.22 \\ \mathrm{P}(\mathrm{E}\mathrm{U}\mathrm{F})=0.85 \end{gathered}$$

Hence, $0.85$ is the probability that the chosen person’s mother tongue is one of Canada’s official languages.